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EXPERIMENTAL RESEARCHES 



THE FLOW OF STEAM THROUGH 
NOZZLES AND ORIFICES 

TO WHICH IS ADDED A NOTE ON 

THE FLOW OF HOT WATER 



BY 

A. RATEAU 

ii 

Engineer of the Mining Corps 
Professeur a VEcole Superieure des Mines de Paris 



'EXTRACT DES ANNALES DES MINES, JANUARY 1902) 



AUTHORIZED TRANSLATION 

BY 

H. BOYD BRYDOIST 




NEW YORK 

D. VAN 1STOSTRAKD COMPANY 

23 Murray and 27 Warken Streets 

1905 





LIBRARY of CONGRESS 
Two Copies Keceivcu 

MAR 3 1905 

Oowngm entry 

€tJt-1-l9o*f- 

CLASS CI AXc. Noi 
COPY B. 






Copyright, 1904 

BY 

D. VAN NOSTKAND COMPANY 



5 



W 



ROBERT DRTTMMOND, PRINTER, NEW YORK. 



PREFACE, 



In the steam-turbine we are confronted with an 
entirely new series of phenomena. The knowledge of 
steam accumulated by familiarity with the reciprocat- 
ing engine is of little use in designing this newest yet 
oldest of prime movers. Nor can the cut-and-try 
methods that have brought the reciprocating engine 
to its present and, it may almost be said, its final 
stage of excellence be employed in the steam-turbine. 
Modern commercial conditions are such that design 
must be based on ascertained physical laws. 

The development of the steam-turbine within the 
last few years has been remarkable. It has reached a 
position not only of equality with the best modern 
engines, but its high efficiency, its lower first cost, 
smaller size, and ease of operation have placed it in the 
very forefront of steam-engineering practice. 

This rapid advance has necessitated the re-investiga- 
tion of steam from the point of view of velocity instead 
of pressure, of kinetic energy instead of potential, of 
the friction of rapidly moving bodies against a vapor 
instead of a lubricated surface, of the balancing of 



1V PREFACE. 

parts so that they shall revolve about their centre of 
gravity, and of numerous other questions which are but 
beginning to make themselves felt. The available data 
are few, and practically all the information is scattered 
widely through the pages of French or German scientific 
periodicals. 

Of all these, the fundamental problem is the propor- 
tioning of the nozzles or passages which shall permit 
the- steam to transform its pressure into vis viva. 
While there exists considerable information on this 
subject throughout the scientific press, very little is 
reliable or in such shape as to be of practical value. 
The experiments undertaken by Prof. Rateau on this 
most important feature of steam-turbine design were 
made when he was beginning his study of the steam- 
turbine and evidence a care and precision of scientific 
research coupled with a keen appreciation of the prac- 
tical that give a report of his results a value to the 
designer too often wanting in the ordinary laboratory 
investigation. 

In the hope that these results may be of some service 
to those who have neither the time nor the inclination 
to study them in the original, this translation has 
been made. 

H. B. B. 

Chicago, September 1904. 



CONTENTS. 



i. 

PAGE 

Outline of the Theory of Steam Flow. Velocity of Discharge. 
Weight of Discharge. Profile of Nozzle for Steam. Prac- 
tical Formula for Discharge 1 

II. 

Experiments. Direct Method of Previous Experimenters. In- 
direct Method of the Author. Estimation of the Precision 
of the Measurements. Arrangement of Apparatus. Re- 
marks on Temperature and Pressure Measurements. Cali- 
bration of the Water-discharge Nozzle. General Discus- 
sion of the Results. Results of the Experiments with the 
Convergent Nozzle. Results Compared with those of Pre- 
vious Experimenters. Experiments with p greater than 
0.58P. Experiments with Orifice in Thin Plate 15 

III. 

Comparison of Results with Hirn's Experiments on Air. Analy- 
sis of Hirn's Results. Discussion of the Curves 43 

Tabulated Results of the Experiments 47 

IV, 

Discussion of the Phenomena of the Flow of Hot Water through 

Nozzles 57 

APPENDIX. 

Bibliography of the Flow of Steam through Nozzles 75 

v 



EXPERIMENTAL RESEARCHES 



ON 



THE FLOW OF STEAM THROUGH CONVERGENT 
NOZZLES AND ORIFICES IN THIN PLATES. 



i. The operation of steam turbines depends upon 
the flow of steam at a high velocity. In order to be 
able to calculate these machines rationally it is neces- 
sary to have a complete knowledge of the phenomenon 
of the flow of steam. As only brief, incomplete and 
sometimes erroneous ideas are foimd in the works on 
mechanics and thermodynamics, I undertook extended 
and precise experimental researches in order to verify 
the thermodynamic principles involved. 

The results of these researches, made in 1895-96, 
are here published. We shall see that the theoretical 
formulae, correctly interpreted, are exactly confirmed, 
and also that the experimental results allow of deter- 
mining very closely the mechanical equivalent of the 
heat unit. 



2 FLOW OF STEAM 

Before describing the method employed I will out- 
line the theory of the phenomenon. 

2. Theory. — The flow of elastic fluids through noz- 
zles differs considerably from the flow of liquids as 
soon as the ratio of the pressure in the exhaust to the 

initial pressure -^- becomes appreciably less than unity. 

If at any point in a nozzle 

S be the area of the nozzle, 

V the velocity of the fluid, 

Q the volume of discharge, 

W the weight discharged, 

D the mean density of the fluid in the state in 
which it exists, i.e., homogeneous or heterogeneous, 
while passing the section S, then the volume of the 
discharge Q is at once given by Q = VS, and the weight 
of the discharge is equal to DQ and is related to the 
section S and the velocity V by the following formula: 

W = DVS (1) 

If D is constant, as in the case of liquids, the section S 
is inversely proportional to the velocity V, and con- 
sequently V always increases as S decreases, but this 
is no longer true in the case of elastic fluids. As the 
pressure falls the density decreases proportionally, 
while the velocity increases in such a manner that 
the product DV first increases, reaches a maximum, 
and then decreases. In the case of gases the maxi- 
mum occurs for a ratio of the pressures equal to 0.52, 



THROUGH CONVERGENT NOZZLES, ETC. 3 

and for steam when the ratio is in the neighborhood 
of 0.58 whatever the initial pressure may be. 1 The 
calculation of this maximum will be given later. It 
follows that when the exhaust pressure is lower than 
0.58P the discharge nozzle, at first convergent, should 
then diverge, if it is desired that the steam shall con- 
tinue to expand so as to reach the speed correspond- 
ing to the fall of pressure from P to p; and the ratio 
of the final section of the nozzle S t to the section at 
the throat S should vary with the ratio of the pres- 
sures. At the throat the pressure is always equal to 
0.58P, and the velocity, which depends on P only, is 
the same as sound would have in the fluid in the state 
in which it exists at that point of the nozzle, as has 
been shown by Hugonoit (Comptes Rendus de FAcade- 
mie des Sciences, vol. 103). 

With a given nozzle, if the pressure p into which 
the steam is discharged is lower than the value p x 

S 
corresponding to the ratio — L of the sections of the 

o 

mouth and the throat of the nozzle, the pressure of 
the steam at the mouth of the nozzle will not be p, but 
p lf which is in a fixed ratio to the initial pressure P. 
At the moment of leaving the nozzle, the steam en- 
tering suddenly into a space where the pressure is lower 
than in the mouth of the nozzle immediately expands 

1 The value of this ratio which makes DV a maximum 
appears, however, to depend a little on the value of P in the 
ease of steam. 



4 FLOW OF STEAM 

and the jet takes a paraboloid form. The enlargement 
of the jet ceases when the back pressure reaches the 
value p v 

The discharge is independent of the back pressure 
when that is less than 0.58P. On the contrary, when 
p is larger than 0.58P the discharge depends both 
on P and p. It is necessary to distinguish two very 
different cases, therefore- in the first the calculation 
of the discharge depends only on P and the formula 
is simple; in the second it depends both on P and 
on p. 

For both cases, however, it is the most contracted 
section of the nozzle (the throat if it is convergent- 
divergent, or the mouth if it is simply convergent), 
which enters into the calculation of the discharge. 

The above is only applicable, strictly speaking, to 
nozzles properly so called. In the case of orifices in 
thin plates the coefficient of contraction K, which 

varies with the ratio -p of the pressures, complicates 

the phenomenon. We shall see later on the experi- 
mental values that were obtained for this coefficient. 

3. Equation of Velocity. — The velocity V of the 
steam can be calculated by two different methods, 
either by the ordinary mechanical method having 
given the relations between the specific volume s and 
the pressure p, or by thermodynamics having given 
the thermal constants of the steam, which can be 
obtained from Regnault's steam tables. 

The first method results in the general formula usu- 



THROUGH CONVERGENT NOZZLES, ETC. 5 

ally credited to Weisbach, but already pointed out 
by Wantzel and Saint Venant in 1839, 

Tg =h-dp,. ..... (2) 



wh'ch can be integrated when the relation between 
v and p is known. 

It is always assumed that in the flow of steam through 
nozzles the expansion is adiabatic because the steam 
remains so short a time in the nozzle, some 0.0001 of 
a second, that it cannot give up or receive an appreci- 
able quantity of heat. Now in the case of adiabatic 
expansion of initially saturated steam Zeuner has 
shown 1 that within large limits of pressure we have 
approximately 

pv = constant, . . . . . (3) 

as in the case of gases, but with ?- = 1.035 instead of 
1.41. Putting the value of v calculated by this equa- 
tion in the preceding equation, V is obtained as a 
function of P and p. Then from Zeuner's empirical 
formula D = 0.587P " 94 , which gives the density of 
steam as a function of its pressure, we have the fol- 
lowing formula for the discharge when P = 0.58P: 

T7 = 15.26P - 97 , (4) 

1 Mechanical Theory of Heat, 



6 FLOW OF STEAM 

W being the weight of steam discharged in grammes 
per second per square centimetre of orifice. 1 

This formula was given by Dr. Grashof (Theore- 
tische Maschinenlehre, Vol. I, in). 

By the second method, developed by Zeuner, the 

V 2 
kinetic energy — of unit weight of the fluid should be 
^9 

equal to the energy represented by the entropy dia- 
gram when the expansion is adiabatic from P to p. 

Putting T the temperature of saturated steam 
corresponding to P, T x that corresponding to p, in 
the entropy diagram (Fig. 1), with entropy <f> as 
abscissae and absolute temperature T as ordinates, AE 
and DF are the isothermals at the temperatures T 
and T v AD is the curve of entropy for water, and 
EF that for saturated steam. Let B be the point 
corresponding to the state of the steam as it enters 
the nozzle. Now the relative weights of liquid and 
vapor in the mixture represented by the point B are 
proportional to the lengths BE and BA. (We are 
here supposing for the sake of greater generality that 
the fluid is not initially in the state of saturated steam.) 
BC is the line representing the expansion of steam in 
the nozzle. If the expansion is adiabatic, BC is a 

1 In English units this formula becomes 
TF = 0.0165P 097 , 

where W = pounds discharged per square inch of orifice per 
second and P = absolute pressure in pounds per square inch. 
—Trans. 



THROUGH CONVERGENT NOZZLES, ETC. 



straight line parallel to the axis of temperatures. 
The total mechanical energy developed by the fluid 
during its complete expansion from the pressure P to 
the pressure p is represented on the diagram by the 
area of the trapezium ABCD. This area is given by 



To 




Fig. 1. 

the simple expression OL, in which 6 is its height (that 

is to say, the fall of temperature from T to T x ) and 

L its mean length MN. 

When the fall of temperature is small this mean 

length L can be taken in practice equal to the mean 

of the two bases AB and CD ; but if the exact formula 

is required, the trapezium must be divided into the 

rectangle ABCG, equal to OAB, and the triangle with 

curvilinear hypotenuse AGD. If the steam is initially 

r 
saturated, AB is equal to 7^7-, r being the heat of va- 

J- 

porization of the water at the temperature T . 



a FLOW OF STEAM 

Introducing E, the mechanical equivalent of heat, 
the formula for the velocity becomes 

V 2 =2gEdL, (5) 

and if the steam is initially saturated, we may write 

F 2 =2^(^-+g) (6) 

C being the specific heat of the liquid and T being 

T -\-T 
nearly the arithmetic mean —^ — - of the extreme 

temperatures. 

This is the simplest form of the equation by which 
the velocity of the flow of steam can be calculated. 
It supposes a knowledge first of the temperatures cor- 
responding to the pressures; second, the heat of vapor- 
ization of the liquid; third, the specific heat of the 
liquid. These values are given for water by Regnault's 
tables with an approximation more than sufficient 
in practice. 

4. Calculation of the Weight of Discharge. — In order 
to calculate the weight of discharge, the velocity of 
flow, the density of the steam at the section of the 
nozzle under consideration, and the quantity of liquid 
formed during the adiabatic expansion must be ascer- 
tained. This liquid is entrained in the steam and 
probably makes with it a homogeneous mixture. 

The density can be obtained from Regnault's tables 
by means of Clapeyron's formula. It has already been 



THROUGH CONVERGENT NOZZLES, ETC. 9 

calculated and is generally found in a supplementary 
column added to Regnault's tables. 

The quantity of liquid formed can be calculated 
from the entropy diagram, or can be taken from a 
chart, as I have shown in my Rapport sur les Turbines 
a Vapeur. 1 The proportion of water thus obtained is 
1 — x, where x is the quality of the steam. 

Neglecting the specific volume of the liquid com- 
pared with that of the steam (which is many hundred 
times larger), we have for the weight of discharge 



w=™v- 



Conversely, being given a weight W of steam discharged 
by a nozzle, we can calculate by this formula the area 
necessary when the pressure has any given value p. 
An example of this calculation will be given later. 

In the first place, however, it should be noted that 
the two methods employed in calculating the velocity 
and weight of discharge while differing in form are 
really identical, because it was from Regnault's tables 
that Zeuner deduced his empirical formula connecting 
the pressure and specific volume of a mixture of steam 
and water during an adiabatic expansion, so that both 
methods are based finally upon Regnault's tables; 
that is to say, on the experimental data of thermody- 
namics. 

1 Memoir presented to the Congres International de Meca- 
nique, 1900. Rev. de Mecanique, August 1900. 



10 FLOW OF STEAM 

5. Profile of Nozzles for Steam. — In the following 
table is given the calculation of the velocity of flow of 
initially saturated steam from an initial pressure P = 10 
kilogrammes per square centimetre. 

The first column of this table gives the pressure p 
at which the speed is to be calculated. 

The second column gives the fall of temperature 6; 
that is to say, the difference of the temperatures cor- 
responding to the pressures P and p. 

The third column gives the value of L ; that is to say, 

r 6 6 

where T' = T- 



T 2T n "— * 2 " 

The fourth column gives the velocity of flow V in metres 
per second calculated from the preceding formula (6). 

In the fifth column the density corresponding to 
the pressure p is given. 

The sixth column contains the product DV. 

In the seventh column is given the proportion of 
steam remaining when the mixture has expanded adia- 
batically to the pressure p. The fraction x is calcu- 
lated by means of the entropy diagram or by Zeuner's 
formula 

r x x x r o x Q n}^„ ^i 

-=— -Clog^. 

In the eighth column the quotient re by DV is written, 
or rather, to avoid a number of zeros, 1000 times this 
quotient, 

1000^. 



THROUGH CONVERGENT NOZZLES, ETC. 



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12 FLOW OF STEAM 

The sections of a nozzle required to discharge a pre- 
determined weight of steam will be proportional to 
this quantity. To obtain the absolute values, the 
actual discharge of steam must be known. 

Examining the numbers in this column it will be 
noted that the areas decrease until p is equal to 0.58P, 
and then increase indefinitely. In order to determine 
as exactly as possible the position of this minimum, we 
have made the intervals of pressure p in the neighbor- 
hood of the value 0.58P much smaller. By tracing 

x 
the curve of the quantity 1000 jyxr~ as a function of the 

pressure p, the position and the value of this minimum 
are determined with an approximation as close as is 
permitted by Regnault's tables. 

I have made this calculation for the discharge from a 
number of initial pressures, and can state that the posi- 
tion of the minimum varies slightly with the pressure 
P around the value 0.58P. 

x 
Taking the square root of the quantity 1000 jy^ and 

forming the ratio of this square root to its smallest 
value, we obtain a quantity which we may call R, which 
will be proportional to the diameters or radii of a noz- 
zle of circular cross-section of proper shape to discharge 
the steam. The values of R are written in the last 
column of the table. Plotting as abscissae either p or 
some function of p, for example log p, and as orclinates 
the value of R, we can represent the longitudinal pro- 
file of nozzles for the flow of steam. Fig. 2 gives this 



THROUGH CONVERGENT NOZZLES, ETC. 



13 



curve by which the nozzles of steam turbines can be 
calculated rapidly for each particular case. 

This curve is utilized in the following manner. Hav- 
ing the initial pressure P and the final pressure p to 
which the steam is required to expand, we take from 





i 






| 


















2 






























*S» 
















































po. 














AXIS 


FTHENOZZL 








( 



4 3 2 1 0.8 0.6 

EXHAUST PRESSURES IN KG. PER CM. 2 . 

Fig. 2. 



Olog P 



the curve the ordinate corresponding to the abscissa in- 

dicated by the value of the ratio j>. This ordinate is 

then the radius of the final section of the nozzle, the 
section at the throat having the minimum ordinate of 
the curve as its radius. 

It is true the curve will be slightly different for differ- 
ent initial pressures, but for the needs of practice the 
variation can be neglected. 

6. Practical Formula for Discharge. — When the nozzle 
is convergent and the final pressure p is less than 0.58P 
the discharge depends only on P, because then the pres- 
sure p at the mouth of the nozzle is always equal to 
0.58P. Calculating the discharge W per unit area of 



14 



FLOW OF STEAM 



the mouth of the nozzle, or calculating the ratio 



W 
P' 



which varies little, we find results such as are indicated 
in the following table. 

W 

Table of the Theoretical Values of —. 



p 


V 


• 


r + 6 
T 2T-0 


V D 


X 


DV 
\0x 


w 
p 


kg/cm 2 

10 

5 

2 

1 


kg/cm 2 
5.8 
2.9 
1.16 
0.58 


degrees 
22.258 
19.337 
16.310 
14.459 


1.0875 
1.2028 
1.3530 
1.4639 


m/sec 1 kg/m 3 
449 .19 3. 0570 
439.85 1.5950 
428.42 0.6738 
420.04 0.3507 
1 


0.9650 
0.9650 
0.9670 
0.9697 


142.30 
72.70 
29.85 
15.19 


14. :3 

14.54 
14.92 
15.19 



w 

The theoretical formula -p- = 15.20 — 0.96 log P gives the same figures as 
in the last volume to within about unity in the last order 

In order to ascertain the relation between the value of 

this ratio and the pressure we may plot as abscissae, not 

lengths proportional to P, but lengths proportional to log. 

W 
P, and as ordinates the values of the ratio -p- obtained 

from Regnault's tables. The points thus obtained 
range themselves almost exactly on a straight line from 
the value P = l kilogramme per centimetre to P = 13 
kilogrammes per square centimetre. The digressions 
do not exceed one-tenth of one per cent, as will be seen 
by the curve on Plate II, where the points calculated 
from theory are marked by dots surrounded by circles. 
We can then write the following very simple empirical 
formula: 

J = a-MogP (7) 



THROUGH CONVERGENT NOZZLES, ETC. 15 

a and 6 being two constants for which the best values 
are a = 15.20 and 6 = 0.96, the discharge being given as 
before in grammes per second per square centimetre 
of the mouth of the nozzle. 
This formula can also be replaced by the following, 

to which it is practically equivalent in practice ( - being 

small) : 

W=aP' - b -\oge, (8) 

e being the base of the natural logs, or in figures 

Tf = 15.20P - 9725 (9) 

This is the form in which the formula is given by Gras- 
hof. (Grashofs formula is T^ = 15.26P°- 9725 when the 
coefficient of discharge is equal to unity. 1 ) 

In practice the first form is the most convenient and 
is the one I have recommended. 

We will see that our experiments lead to slightly 
increasing the coefficient 15.20 and bring it to exactly 
the value 15.26 adopted by Grashof. 

EXPERIMENTS. 

7. Previous Experiments. — Previous to the work 
which I now publish and of which the complete re- 

1 In certain German works a formula is given as Grashofs 
in which the coefficient is a little smaller than 15.26 (by about 
1%) in order to allow for the contraction of the jet. 



1 6 FLOW OF STEAM 

suits are given, we possessed several series of experi- 
ments on the flow of steam; especially in France those 
of Minary and R6sal in 1861 ; x and more recently 
those of M. H. Parenty in 1891 2 ; in America those of 
Napier in 1866 3 ; then those of Peabody and Kun- 
hard in 1888 and 1890/ and Miller and Read in 1895 5 ; 
in England the experiments made quite recently by 
Rosenhain in 1899. 6 All these authors have used the 
same method, consisting in condensing the steam after 
it has passed through the nozzle under test in a surface 
condenser, and then weighing the water thus condensed 
during a certain lapse of time. 

This method, which may be called the direct method, 
presents many inconveniences, however. 

Firstly: One cannot operate on large nozzles without 
difficulty because a very large condenser is necessary 
for the experiments, and it is desirable to employ large 
nozzles — first, because then we approach the condi- 
tions of practice, and, second, the effect of friction 
and the thermal influence of the walls are reduced. 

Secondly: The duration of a series of experiments is 
unavoidably long, since for each measurement it is 
necessary to wait until a considerable quantity of water 



1 Annates des Mines, 5th serie, Vol. IX. 

2 Annates de Chimie et de Physique, May 1896. 

3 On the Velocity of Steam and Other Gases. Spon, 1896. 

4 American Society of Mech. Eng., 1890. 

5 Technology Quarterly of Boston, 1895, Vol. VIII. 

6 Experiments on Steam-jets (Pro. of Civil Eng., 1900). 



THROUGH CONVERGENT NOZZLES, ETC. 17 

has been condensed, which involves each trial lasting 
ten to fifteen minutes. 

Thirdly: The method is not very precise because of 
the incessant variation of the steam-pressure. It is 
'very difficult, in fact, to maintain the pressure of a 
boiler practically constant during some ten minutes, 
and the variations are reproduced at the nozzle. This 
involves taking averages, which complicates the experi- 
ment and renders it less precise. 

Fourthly and finally: The entrained moisture is 
weighed in addition to the steam discharged, and con- 
sequently the measurement is falsified by a correspond- 
ing quantity nearly equal to the proportion of water, 
so that results obtained by this method are generally 
too large. For example, on the diagram in Plate II 
some of ResaFs results are plotted (by triangles), and 
it will be seen that they are about 3 per cent too 
high. Those of Rosenhain and other authors give 
similar results. 

8. Actual Experiments. — To avoid these sources of error 
all readings should be completed in a very short interval 
of time, as soon as the steady state of flow is properly 
established. The experiments can then be made rapidly 
and precisely. I achieved this result by the application 
of a new and indirect method consisting in condensing 
the steam discharged by the nozzle in a current of 
very cold water and at the same instant measuring the 
total discharge of water and the rise in its temperature. 
This method is founded on the use of an ejector con- 
denser. The total quantity of water flowing after con- 



1 8 FLOW OF STEAM 

densing the steam was ascertained by discharging the 
water through a conical nozzle under a head which 
could be accurately measured. Knowing the total 
quantity discharged Q and the rise of temperature, the 
weight of steam discharged X is deduced from the 
specific heat C and the total heat of formation of steam 
X by the formula 

C8 

x ^i^cr a ^ 

t Q being the initial temperature of the water. 

The advantages of this method are evident. The 
duration of each individual test did not exceed two to 
three minutes. The steady state was established in 
about a minute, and when all the instruments had 
reached a stable condition the reading of their indica- 
tions was made in a few seconds. The conditions of 
pressure were then changed, and so on. The accuracy 
of the experiments I estimate as follows: 

9. Estimation of the Precision of the Measurements. 
— The increase of temperature was on the average 
about 20° C, and the measurement of each of the initial 
and final temperatures in the ejector condenser were 
made to within one twentieth of a degree, which gave 
a mean error of 1 in 400. As I shall show later, how- 
ever, a cause of error was always present which was 
quite important; namely, the displacement of the zero 
of the thermometer on the discharge side of the con- 
denser during the course of the measurements. This 
displacement reached about two tenths of a degree, 



THROUGH CONVERGENT NOZZLES, ETC. 19 

being nearly 1 per cent, or one half of 1 per cent, above 
or below the mean. The initial and final pressures were 
obtained by different gauges. At first metallic gauges 
were used and afterwards a mercury manometer pro- 
vided with three-way cocks which allowed of the 
branches being placed in communication either with 
the atmosphere or with the interior of the pipes, when 
the pressure or the difference of pressures did not 
exceed 1 kilogramme per square centimetre. I was 
thus able to measure by this mercurial manometer 
either the initial or final pressures or their difference. 
The principal reading, that of the initial pressure P, 
was ordinarily made by means of a large Bourdon 
gauge especially constructed for my experiments and 
reading to T -J~o of a kilogramme, which for 5 kilo- 
grammes mean pressure corresponds to a mean ap- 
proximation of 0.2 per cent. The mean head H on 
the nozzle for measuring the water discharged was 
about 450 millimetres. As the error of reading was 
less than \ millimetre, the error in calculating the 
discharge could not exceed an average of one in 1800. 
The coefficient of discharge of the nozzle was pre- 
viously tested. The error which could enter into 
this determination, which will be described later on, 
was a few thousandths at the most. 

The diameter of the steam-nozzles was obtained with 
great precision by means of mandrels calibrated to T ^ 
of a millimetre. As the mean diameter was 15 milli- 
metres, the error in the diameter could attain 1 in 1500, 
or in the area 1 in 750. 



20 FLOW OF STEAM 

A cause of error which was very difficult to eliminate 
was the expansion of the nozzles while the steam was 
passing through them. While the coefficient of expan- 
sion of the bronze was known to be 1 in 600, the tem- 
perature at the end of the nozzle could not be ascer- 
tained exactly, but it could be estimated certainly to 
within 40 degrees, and this was sufficiently close, be- 
cause the resulting error was then less than 1 in 1500 
in the diameter, or 1 in 750 in the area. 

To sum up, then, all the individual errors of the 
different readings, that due to the displacement of the 
zero of the thermometer at the discharge excepted, are 
of the order of one to three one-thousandths. Added 
together they would give a total accidental error in the 
calculated results exceeding 1 per cent, but taking one 
with another, the mean error would be probably not 
more than a few thousandths. 

Two sources of error must be examined which are 
inseparable from the method adopted: firstly, the 
entrained moisture, and, secondly, the effect of radia- 
tion from the apparatus. 

Regarding the entrained water it should be remarked 
that our method avoided this cause of error as far as 
possible. Because if there were, for example, 1 per 
cent of entrained moisture, this quantity would tend 
to reduce slightly the velocity of discharge of the steam 
passing through the orifice, and consequently to dimin- 
ish the total quantity flowing per second. But, on the 
other hand, some of the heat contained in this water 
will be given up to the water leaving the condenser 



THROUGH CONVERGENT NOZZLES, ETC. 21 

and replace part of the heat lost by the reduction of the 
flow of steam; so that, finally, if the calculation be 
made, it is found that the actual deficit of heat will 
be only about half that corresponding to the propor- 
tion of water. This method, then, even if it does not 
eliminate the influence of the entrained water com- 
pletely, reduces that influence by one half; and further, 
the error is in the opposite direction to that produced 
in the surface-condenser method. In place of adding 
to the discharge it decreases it. 

But I arranged to avoid entrained moisture as far 
as possible, and also to measure its amount. A sepa- 
rator was placed in the steam-supply pipe just before 
the nozzle, and between this separator and the nozzle 
a sample of the steam was taken which was analyzed 
in an apparatus especially designed for this object, 
which permitted the moisture in the steam to be meas- 
ured. In practice I never had more than two to three 
one-thousandths of water in the steam after passing the 
separator, so that the error in the measurements caused 
by the entrainment of water should be less than fifteen 
ten-thousandths. 

As to the effect of radiation, though the apparatus 
was not lagged, this was relatively very small because 
of the large discharge of steam. In order to ascertain 
the importance of the loss of heat by radiation, I passed 
a small current of steam at atmospheric pressure through 
the apparatus and weighed the water condensed by 
the cooling action of the walls. The quantity thus 
obtained was 2.3 kilogrammes per hour. As the mean 



22 FLOW OF STEAM 

temperature of the parts exposed to radiation between 
the steam-nozzle and the discharge thermometer was 
generally much less than 100° C. during the experi- 
ments, it may be admitted that the radiation was less 
than 0.6 kilogrammes per horn, while the discharge 
was on the average 500 kilogrammes, or a loss of twelve 
ten-thousandths at the most. Finally, it may be 
observed also that the influence of radiation has the 
same effect as the entrained moisture and could cause 
only a deficit in the calculated quantity of steam, 
while we shall see that the experimental results give 
somewhat larger figures for the discharge than those 
calculated theoretically from Regnault's tables. 

A difficulty arose in measuring the water discharged 
on account of the air contained in it. This air, at first 
dissolved in the water, is set free under the influence 
of the vacuum in the condenser and remains in this 
state up to the nozzle used for measuring the water, 
so that if precautions were not taken the discharge 
from this nozzle would be a mixture of water and air. 
Under these conditions measurements of the quantity 
of water discharged would have been erroneous. This 
difficulty was overcome as shown in Plate I by arrang- 
ing a separator in the tank below the water-nozzle by 
means of which the air was disengaged and carried to 
the upper portion of the tank, where it was discharged 
through a small nozzle three millimetres in diameter, 
which permitted of its being measured. The water 
discharged by the principal nozzle was then quite clear 
and free from air. 



THROUGH CONVERGENT NOZZLES, ETC. 23 

10. Arrangement of the Apparatus. — The experi- 
ments were made during the winter of 1895-96 at St. 
Etienne, in the factory of Bietrix & Company, who 
obligingly placed at my disposal a boiler capable of 
giving 1000 kilogrammes of steam per hour at a pres- 
sure of 15 kilogrammes per square centimetre. The 
experiments were made in winter so as to be able to 
use cold water at a temperature of about 6° C. and ob- 
tain a rise of 40° to 50° C. 

Three convergent nozzles were tested successively, 
then an orifice in a thin plate; finally we tested a con- 
vergent-divergent nozzle. The convergent nozzles are 
shown in Plate II. They had a final diameter of 10.49 
millimetres, 16 millimetres, and 24.20 millimetres respect- 
ively. The thin-plate orifice, also shown in Plate* II, 
had a diameter of 20.12 millimetres. 

In order to make all the necessary readings the 
arrangement of apparatus was somewhat complicated. 
It is represented in perspective at the left and diagram- 
matically at the right of Plate I. 

T is the nozzle under test arranged in the joint / be- 
tween the steam-supply pipe N and the exhaust-pipe B. 

The supply-pipe N had an inside diameter of 60 
millimetres. The exhaust-pipe D (inside diameter 120 
millimetres) was fitted with a plate p forming a screen 
for the current of steam leaving the nozzle so as to 
break up the jet. 

A is the ejector condenser. The condensing water 
enters by the pipe C and leaves by the pipe D. 

S is the water-nozzle of the condenser. The distance 



24 FLOW OF STEAM 

between it and the throat of the condenser could be 
varied by the handle g. By means of this nozzle the 
current of steam entering the condenser could be throt- 
tled. 

K is a cock on the steam-supply pipe operated by 
the handle 18. This cock enabled the initial pressure 
at the nozzle to be varied at will, while the back pres- 
sure could be varied similarly by the condenser-nozzle 
S. 

J is a centrifugal separator. The separated water 
is discharged by the cock 1 and the pipe 2. 

L is the apparatus for measuring the moisture remain- 
ing in the steam. The description of this apparatus 
is given in Annales des Mines. 1 It is composed of a 
coil 4 heated by a flame 5, and a combining-chamber 6. 
The sample of steam obtained by the cock 3 is separated 
into two equal portions. One half goes directly to 
the combining-chamber, while the other is superheated 
in the coil 4. The mixture is discharged by the regulat- 
ing angle-valve 9. 

The superheat in the half passing through the coil 4 
is measured by the thermometer t 3 , and the superheat 
of the mixture by the thermometer t 2 . The pressure in 
the chamber is observed by the gauge M 3 . The moist- 
ure in the steam is obtained from the readings of these 
two thermometers and the gauge. 

E is a sheet-iron tank in which the air is separated 

1 A.. Rateau. Appareils servant a mesurer Thumidite d'une 
vapeur. Annales des Mines, April 1897. 



THROUGH CONVERGENT NOZZLES, ETC. 25 

from the hot water leaving the condenser. This tank 
is divided about the middle by a semicircular baffle 13 
under which the air collects. The bubbles rise to the 
upper part of the tank by the pipe 14, and the air flows 
into the atmosphere by the small nozzle /. The pres- 
sure of discharge of the air is measured by the water- 
gauge 19. 

F is the nozzle for measuring the discharge of water. 
It has a final area of 10.99 square centimetres at 15° C. 
The head under which the flow takes place is measured 
by the water-gauge hh. Adjacent to this gauge is a 
thermometer t 5 by which the temperature of the gauge 
could be measured and the density of the water de- 
duced. 

G is a barrel having a capacity of 350 litres into which 
the water is discharged by the nozzle and serves to 
calibrate the nozzle. The barrel is provided with a 
glass tube KK, allowing the exact level of the water to 
be ascertained. Further, a baffle 17, arranged in the 
arc of a circle, receives the jet of water in a thin film so 
that it falls into the barrel without shock and without 
disturbing the surface of the water. 

M lf M 2 , M 3 are the pressure-gauges for measuring 
the initial and final pressures at the steam-nozzle T. 

The mercury manometer is shown at. cd. One of the 
branches c communicates with either the high-pressure 
side of the nozzle or the atmosphere by means of the 
three-way cock 11; the other branch d communicates 
with either the atmosphere or with the discharge side 
of the nozzle by the three-way cock 10. By means 



26 FLOW OF STEAM 

of these three-way cocks the initial or final pressures 
or the difference between them could be measured as 
the case required. In addition the barometric height 
and the temperature of the surrounding air were care- 
fully observed so as to obtain the absolute value of 
the readings of the manometer. 

The temperatures t and t x of the current of water before 
and after passing through the condenser were measured 
by the thermometers e and /. These mercury ther- 
mometers were graduated to twentieths of a degree 
and were previously compared very carefully with a 
Baudin standard thermometer. I also reduced the 
scale to that of the air-thermometer, so that with the 
exception of the displacement of the zero, regarding 
which I will speak immediately, all temperatures are 
given to within one twentieth of a degree less, or one 
tenth of a degree more, than the scale of the air-ther- 
mometer. 

ii. Remarks. — Some observations are necessary on 
the subject of measuring the principal temperatures 
and pressures. 

The reading of the pressure-gauges presented no 
difficulty. For pressures below 10 kilogrammes per 
square centimetre I employed the standard gauge M v 
and for pressures above this the standard gauge M 2 . 
But the management of the mercury manometer offered 
some difficulties. The steam condensed in the tubes 
above the mercury and formed columns of water often 
broken up with air-bubbles, and the reading of the 
manometer had to be corrected for these broken col- 



THROUGH CONVERGENT NOZZLES, ETC. 27 

unins of water. On account of this the readings of the 
mercury manometer were not always as exact as I 
could have wished. When the columns of water were 
not discontinuous, as sometimes happened, the cor- 
rection could be made with certainty. 

The thermometers e and / were plunged in small iro 1 
wells penetrating into the interior of the pipes convey- 
ing the water. These thermometer-cups were filled 
with mercury, so that the transmission of heat was 
very rapid. In fact the reading of the thermometers 
became steady in a few seconds. It may be asked 
whether these thermometers really indicated the true 
mean temperatures of the water. In the case of ther- 
mometer e, placed in the current of cold water, there 
can be no doubt that this was so, for the temperature of 
the cold water varied very little. But in the case of 
thermometer /, plunged in a current of water which was 
necessarily but little homogeneous, immediately at the 
outlet from the ejector, this was more doubtful, especially 
as the temperature of this water varied greatly from 
one experiment to another. I think, however, that the 
readings of these temperatures were always very close 
to the true mean, because on putting a thermometer 
into the water at the mouth of the nozzle F, where 
the temperature had evidently become uniform through- 
out the mass of water, I found that this thermometer 
always gave the same reading as thermometer / to 
within one tenth of a degree, sometimes more, some- 
times less. This difference may be attributed to 
the walls of the tank E not immediately reach- 



28 FLOW OF STEAM 

ing their steady temperature after a change of con- 
ditions. 

The thermometers used were made of ordinary glass. 
I had them carefully compared, as I said, with a stand- 
ard Baudin thermometer to obtain the necessary cor- 
rections, if possible, to one twentieth of one degree. 
Unfortunately I found out too late that I had been 
wrong in not using a good thermometer made of 
hard glass for measuring the temperature of the hot 
water. I thought that up to 50° C. the displacement 
of the zero occasioned by the expansion of the glass 
could be neglected. I found that it was not so, how- 
ever. 

My first calculations having disclosed an error of 
more than 1 per cent between the experimental results 
and the theoretical figures, I suspected that the ther- 
mometer was incorrect, and standardized it again de 
novo. I found that its zero was displaced fully two 
tenths of a degree when it was subjected to a tempera- 
ture of 50°, which was sometimes exceeded in my ex- 
periments, and that the zero returned to its original 
position when cold. This annoying variation in the 
zero was sufficient to render some of my results incor- 
rect by over 1 per cent. 

To reduce this influence to a minimum I ascertained 
the mean position of the zero due to the changes of 
temperature during several successive days. Finding 
that this mean differed by 0.15° from that which I had 
first obtained, I corrected all my results by this amount. 
For this reason the chart and the conclusions given 



THROUGH CONVERGENT NOZZLES, ETC. 29 

here for the formula for the discharge are not identical 
with those in my Rapport sur les Turbines a Vapeur 
au Congres de Mecanique Applique, 1890. 1 

12. Calibration of the Water-nozzle. — The deter- 
mination of the total weight of water discharged de- 
pends upon the coefficient of discharge of the water- 
nozzle. Consequently it was necessary to measure this 
coefficient with the utmost possible accuracy at the 
outset. This calibration was repeated several times. 
The diameter of the mouth of the nozzle was 37.4 
millimetres at 15° C. 

Two glass reference-marks were fixed in the barrel 
into which the water was discharged by the nozzle, 
and the water contained in the barrel between these 
two points was carefully weighed. At an interval of 
two months the weights obtained were 278.2 kilo- 
grammes and 278.05 kilogrammes at about 15° C. ; so 
that the error to be allowed for in this measurement was 
not more than one in two thousand. 

While the head H measured by the water-gauge hh 
remained quite constant, the time required to fill the 
barrel between these two marks was noted with a 
good seconds chronometer. To give an idea of the 
approximation obtained I produce below the calibra- 
tion measurements made on the 27th and 30th of Janu- 
ary, 1896. (See the Table below.) 

1 This report was drawn up hurriedly a few days before the 
Congress. The portion relating to the flow of steam contains 
some errors which are rectified here. 



3o 



FLOW OF STEAM 



EXAMPLES OF THE CALIBRATION OF THE WATER- 
NOZZLE. 
Diameter of the nozzle 37 A mm. Contents of tank 278. 5 litres. 



Temperature 
of the Water. 


Head, 
H. 


Tempera- 
ture t of 

the Water- 
manom- 
eter. 


Time to 
Fill Tank. 


Time 
reduced to Averae . es 
# = 478 A ^erages. 

mm. 


Co- 
efficient 
of Dis- 
charge. 




Millirn. 




Seconds. 


Seconds. Seconds. 




^ f 29.35° 


549 


22° C. 


79.0 


84.6 I 

86:2 h 85 - 2 




N g 125.70 
§2 1 31.90 


560 

581 


17 
17 


79.6 

77.5 


0.9711 


" [22.65 


562 


18 


78.2 


84.8 j 




© [32 
M § 130 
§2? 1 29 


478 


19 


84.7 


84.7 1 






497 
466 


23 
16 


83.0 
86.0 


84.7 
84.9 




•84.75 


0.9763 


^ [29 


451 


21 


87.2 


84.6 









As will be seen, the results differ by a few thou- 
sandths. Regarding the management of the chronom- 
eter, it will be understood that it was difficult to 
avoid errors of \ second which correspond to a relative 
error of three thousandths in a duration of about 80 
seconds. 

In calculating the discharge, I adopted the mean 
coefficient 0.9750, which is perhaps a little high. It 
is astonishing that the coefficient of discharge reached 
a value so close to unity; but that was the result of 
the experiments, and it was necessary to take the coeffi 
cient obtained. 

I regret very much that I did not replace the 350- 
litre barrel by a much larger tank, say of 3 or 4 cubic 
metres capacity, so that the discharge could have been 
measured for some ten minutes. I would then have 



THROUGH CONVERGENT NOZZLES, ETC. 5 * 

obtained an approximation certainly to one one-thou- 
sandth. 

13. Results of the Experiments. — Three convergent 
nozzles were tested successively, and then an orifice in 
a thin plate. The nozzles and the orifice, which were 
of bronze, are represented to scale in Plate II. The 
convergent nozzles had, as I have already said, a diam- 
eter at the mouth of 10.49 millimetres, 15.19 milli- 
metres, and 24.20 millimetres, respectively, at 15° C. 
The thin-plate orifice had a diameter of 20.12 milli- 
metres at 15° C. These diameters were measured with 
great precision to 0.01 millimetre, and the orifices were 
previously corrected by passing calibrated mandrels 
through them so as to render them as perfectly circular 
as possible. 

The results of the experiments and the calculations are 
given in the tables attached to this paper. The ex- 
periments are numbered 1 to 152 ; eleven have not been 
transcribed because they appear to me to contain very 
large accidental errors. These are numbered 9, 38, 56, 
57, 58, 61, 82, 83, 84, 91, and 92; number 47 may also 
be omitted. 

The first column of the tables gives the number of 
the experiment. Columns 2 and 3 are the absolute 
initial and back pressures. These figures are corrected 
from the observed readings and represent the true 
values. For example, a small quantity, about 0.01 
kilogramme, due to the velocity of the steam in the 
supply-pipe, has been added to the initial pressure as 
indicated by the gauge. The gauge reading is due to 



32 FLOW OF STEAM 

the static pressure only, while the true initial pressure 
on the nozzle is the sum of the dynamic pressure and the 
static pressure. 

Column 5 gives the head H on the axis of the water- 
nozzle, 1 and the figures correspond to the temperature 
of the water flowing through the nozzle. 

Columns 6 and 7 indicate the temperatures t and 
t t before and after the ejector condenser. As I have 
explained above, these are reduced to the readings of 
the air-thermometer. The difference between these 
figures gives the rise in temperature t t — 1 of the water 
(column 8). 

The volume of water discharged was obtained by 
the formula 



Q=KSV2gH, (11) 

where Q = cubic centimetres per second, 

£ = area of the nozzle in square centimetres, 
i£ = coefficient of discharge = 0.975, 
i7 = head in centimetres. 
The area S of the nozzle is corrected for expansion 
at the temperature of the current of hot water. The 
weight discharged in kilogrammes per second is written 
in column 9. 

The quantity of steam flowing per second is then 
calculated by means of the total discharge of water 
and from the temperatures t and t v Calling X the 
total heat of the steam above 0° C, and C the mean 

1 This was a brass nozzle. 



THROUGH CONVERGENT NOZZLES, ETC. 3$ 

specific heat of the water between the temperatures 
t and t t1 X the weight of steam discharged per second 
by the nozzle, and Q—X the quantity of injection water 
supplied to the condenser, the water resulting from 
the condensation of the steam being at the tempera- 
ture t ly then the quantity of heat given up by the 
steam 1 H = X(X—Ct 1 ), while the quantity of water 
Q — X is heated from ^to^and absorbs (Q—X)C(t l —t ) 
heat units. We have then the following equation: 

whence 

where d=t 1 —t . 

The quantity of heat A is given by the formula or 
by Regnault's Tables, and the specific heat C of the 
water is given by the well-known expression 

£=1 + 0.4^-0.9^. 

The steam discharged, calculated in this manner, 
expressed in grammes per second, is given in column 10 
of the tables. 

Dividing by the area of the nozzle, we have the dis- 
charge per square centimetre in column 11. 

1 During the expansion of the steam in the nozzle a certain 
quantity of heat is transformed into mechanical energy, but 
this quantity is immediately and entirely given back again 
by the destruction of the kinetic energy of the jet, so that 
it is quite correct to take X(X — CQ as the quantity of heat 
given up by the steam. 



34 FLOW OF STEAM 

A remark is necessary here. What value should be 
taken for the area of a nozzle? The diameter I have 
given is measured cold (15° C), while under the action 
of the current of steam the nozzle becomes heated and 
expands. It is impossible to know exactly what to 
take for the temperature of the walls, because the cur- 
rent of steam has a varying pressure and also a varying 
temperature during its passage through the nozzle; 
and further, the walls are subjected to different tem- 
peratures at different points. It seems to me to be 
useless to try to calculate the true temperature of 
the nozzle in each experiment. 

I adopted finally a mean temperature of 120° C. and 
based the calculations on the diameters corresponding 
to this temperature. This is one of the difficulties of 
these researches on the flow of steam and also on the 
flow of gases. To eliminate this cause of error entirely, 
it would be advantageous to employ metals having a 
very small coefficient of expansion, such as the new 
alloys of iron and nickel. When my experiments were 
made, however, these alloys were not yet known. It 
is unnecessary to give too much importance to this 
cause of error. The variation of temperature of the 
nozzle did not differ by more than about 40° C. from 
the mean temperature adopted (120° C), and for bronze 
40° corresponds to a linear expansion of one in fifteen 
hundred, so that the error in the area did not exceed 1 
in 750, or but little more than one tenth of one per cent. 

Finally, the last column of the table gives the ratio 
of the discharge of steam IT' to the initial pressure P 



THROUGH CONVERGENT NOZZLES, ETC. 35 

when p is less than 0.58P; or the ratio of the actual 
discharge W to the maximum discharge W m when the 

ratio -p is greater than 0.58. These two cases are 

always distinguished in the tables, because in the first 

case the discharge depends on the initial pressure P 

which is constant (as is p), while in the second case it 

depends on the back pressure also. 

We will now examine the results of the experiments 

and compare them with those given by theory. 

W 
14. Convergent Nozzles. — The ratio -5- varies but 

slightly. It oscillates about the value 15. A very 
clear chart can be made by plotting as ordinates the 
value of this ratio, as is done on Plate II, where I have 
taken as abscissae the logarithm of the initial pressure. 
This graphic representation is very satisfactory because 
the origin of the coordinates is a long way off the 
paper, so that variations of only 1 per cent correspond 
to a relatively large height. To give an idea of the 
amplification of the ordinates, I have drawn the lines 
ab and a'b f corresponding to a difference of 1 per cent 
above and below the theoretical line AB. It will be 
seen that the majority of the points lie above the 
theoretic line AB. Only a few points are more than 
2 per cent away from this line. If the mean distance 
of the experimental points from the theoretical curve 
be taken for each nozzle, the following results are 
found: 
For the nozzle 10.49 millimetres diameter (21 



36 FLOW OF STEAM 

experiments) the mean departure is 1.17 per cent 
high. 

For the nozzle 15.19 millimetres diameter (19 ex- 
periments) the mean departure is 0.69 per cent high. 

For the nozzle 24.20 millimetres diameter (19 ex- 
periments) the mean departure is 0.25 per cent high. 

These are very small differences and clearly show the 
satisfactory agreement between the experiments and 
theory. 

If the curve for the mean results of the experiments 
be plotted, it will be seen that it follows the direction of 
the theoretical curve closely, but is a few thousandths 
above it. This difference between experiment and 
theory can be accounted for by several causes. 

First of all, in the theoretical calculations I have 
taken the mechanical equivalent of the calorie as 425. 
This is a little low, as is now recognized. Now E* enters 
into the expression for the speed of the steam. If 428 
be taken instead of 425, the figures for the theoretical 
discharge will be increased by 0.35 per cent, and the 
mean difference between experiment and theory will 
not be more than 0.35 per cent. 

Further, we have seen that the calibration of the 
water-nozzle was not as accurate as could have been 
wished; perhaps a certain error is to be feared there. 
I do not believe that it exceeds 0.3 per cent. 

Finally there was a displacement of the zero of the 
thermometer measuring the temperature of the hot 
water. This is certainly the most important source of 
error, I stated above that I was led to modify the 



THROUGH CONVERGENT NOZZLES, ETC. 37 

first standardization by 0.15° after having carefully 
observed the variations of the thermometer. Never- 
theless it may be that the mean position of the zero dur- 
ing the experiments was slightly different from that 
which I finally adopted. An error of less than one 
tenth of a degree would be sufficient to account for 
the remaining error of 0.35 per cent. 

I have indicated on the chart the points correspond- 
ing to the old experiments by Minary and Resal. These 
points are surrounded by a triangle; one of them has 
not been plotted, as it is outside the limits of the figure. 
It appears that these experiments gave very much 
larger results than ours, which is explained, as I have 
already remarked, by the method employed not al- 
lowing of a correction being made for the effect of the 
entrained moisture. The error reaches 5.2 per cent 
and averages 2.5 per cent. 

Similarly Rosenhain's experiments on nozzle No. 4 
give an error reaching 3 to 4 per cent. 

We shall see shortly in analyzing Hirn's experiments 
on the flow of air that he had the same trouble with 
that fluid. 

Thus there is a satisfactory agreement between ex- 
periment and theory, but the experimental discharge 
appears to be a little larger than what we should expect 
from theory. 

Should the theoretical formula be followed in practice, 
or should we adopt the coefficients given by the pre- 
ceding experiments? If it be desired to use the theo- 
retical formula, the coefficient must be taken as 15.20 



3& FLOW OF STEAM 

for £ = 425, or 15.25 for # = 428; and if it be desired 
to follow experiment, a value of 15.32 at most should 
be taken for the coefficient. The practical formula 
then will be 

-^ = 15.20 to 15.32-0.96 log P, . . (13) 

within the limits of our experiments; that is to say, 
between P = 1 and P = 12 kg. per square centimetre. 

In my report to the Congres de Mecanique Applique 
I gave 15.42, which resulted from a summary glance 
over the experiments, as the first term of the second 
member of this equation, and for the second term 
— log P. At the International Congress in Glasgow I 
suggested the value 15.20 resulting from the theoreti- 
cal formula in which E is taken as 425. These are the 
limiting values of this term. I estimate that the most 
probable actual value is 15.26 with a possible error 
less than 0.4 per cent. This is exactly the coefficient 
in Grashof s formula which was deduced from theory 
by the first method given on page 15. Grashofs 
formula, however, gives a too rapid decrease for the 

W 

ratio p-, as will be seen from the figure on Plate II, where 

the line CD corresponds to this formula. 

The value of the coefficient of log P is between 0.96 
and unity. At first I always took imity because of 
the simplification which results; nevertheless it is 
better to take 0.96, which is probably more exact be- 
cause it is given by the thermodynamic calculation, 



THROUGH CONVERGENT NOZZLES, ETC. 39 

15. Remarks. — The agreement between experiment 
and theory to less than 1 per cent leads to the conclu- 
sion that, contrary to the opinion sometimes stated, 
there is no sensible retardation in the condensation of 
the steam. In fact the calculation takes into account 
the condensation during the expansion, and the pro- 
portion of steam condensed when the pressure has 
become equal to 0.58P is about 0.3 to 0.36 per cent, 
depending on the value of P. Now if any retardation 
in the condensation occurred, experiment would reveal 
a deficit amounting to 0.3 per cent to 0.35 per cent in 
the quantity of steam discharged. But in place of 
a deficit, a too high value is always found so that it 
may be concluded that if there is any retardation 
of the condensation it is very small, of the order of 
0.00001 of a second. 

On the other hand it will be seen also that the co- 
efficient of discharge for convergent nozzles of the form 
tested by me is very close to unity for values of p less 
than 0.58P, since the experimental discharge is always 
found to be larger than the theoretical. A similar con- 
clusion results from Hirn's experiments on air. When 
p is greater than 0.58P this is not the case, how- 
ever. 

Finally, if, conversely, we suppose a priori that the 
coefficient of discharge is unity, then the value of E, 
the mechanical equivalent of the calorie which will make 
the experimental discharge equal to the theoretical, 
is found to be 431, which differs by 0.7 per cent only 
from the value 428 generally adopted. 



4° FLOW OF STEAM 

16. Experiments with P larger than 0.58P. — When 
the back pressure is larger than 0.58P the flow depends 
not only on the initial pressure P, but also on the back 
pressure p. The results can be best represented by- 
taking the ratio of the actual discharge W to the maxi- 
mum discharge W m , which occurs when the back pressure 

W 

is less than 0.58P. This ratio -^ — is then practically 

yy m 

v 

independent of the ratio -p of the pressures. The 

points can be plotted on cross-section paper and the curve 

W v 

of yrj— as a function of -^ constructed. 

VV m ± 

For the experiments where p was larger than 0.58P 
the tables give, in column 12, the theoretical maximum 
discharge W m calculated by the formula given above: 



W m =P(15.2G- 0.96 log P), . . . (14) 

W 
and in column 13 the ratio ^-. 

rr tn 

The results of these experiments have been plotted 
in this manner on Plate III, the results for each of the 
three convergent nozzles experimented on being differ- 
ently indicated. 

It will be seen that the points fall very well on a 
curve of the elliptic form, as was pointed out by M. H. 
Parenty. 1 As I have remarked above, the larger de- 



1 Annales de Chimie et de Physique, May 1896. 



THROUGH CONVERGENT NOZZLES, ETC. 41 

partures from this curve may be considered as being 
due to the great uncertainty that sometimes occurred 
in measuring the difference of the pressures by the 
mercury manometer when the column of water sur- 
mounting the column of mercury was interrupted by 
bubbles of air. The standard gauges also gave but a 
poor approximation at low pressures» It will be no- 
ticed further that the greatest differences occur in the re- 
sults for the largest nozzle 24.20 millimetres in diam- 
eter. The results given by the small nozzle are much 
more regular. 

To compare the results obtained with theory the 
curve AB, based on the thermodynamic calculation, 
has been drawn. This curve is easily obtained by 

DV 

taking the ratios of the function defined on page 9 

x 

to the maximum value of this function. It is simply 
necessary to take the reciprocal of the ratios of the 
numbers in column 8 of the table on page 11 to the 
smallest of these numbers, 0.70276. 

It will be noticed that the experimental curve lies a 
little below the theoretical curve, as would be expected, 
and that it becomes horizontal a little later than the 
theoretical curve, which means that the maximum 
experimental discharge is not attained until the back 
pressure is lower than 0.58P. By taking the ratio 
between the ordinates of the experimental and theo- 
retical curves the coefficient of discharge for the con- 
verging nozzles is obtained, from which it will be seen 
that this coefficient starts with a value about 0.94 



42 FLOW OF STEAM 

when the ratio of the pressures is in the neighborhood 

of unity and gradually increases towards the value 

unity. As the experimental curve and also the theo- 

" W 
retical curve are very nearly ellipses, the ratio ^- as 

"TO 

a function of the ratio -p of the pressures can be ex- 
pressed by a very simple formula, but it does not seem 
worth while to go into this. 

We shall see that the curves for air are also very 
nearly quadrants of an ellipse. 

17. Orifice in a Thin Plate. — The orifice in a thin 
plate does not behave in the same way as the convergent 
nozzles. The discharge does not reach a maximum for 
p equal to or a little less than Q.58P, but increases 
constantly as p falls. 

The experimental figures are given in the tables 

W 
either with the ratio -5- or, which is better, the ratio 

W_ 
W m ' 
The results relating to this orifice are also plotted 

graphically on Plate III up to the value of -p equal to 0.4. 

It will be noticed that the points fall very regularly on a 
curve EF which is always rising. If the ratio between 
the ordinates of this curve to those of the theoretical 
curve AB be taken, it will be seen that the coefficient of 
discharge for the thin-plate orifice starts at a value of 
0.61 for small differences of pressure and increases 



THROUGH CONVERGENT NOZZLES, ETC. 43 

very regularly to the value 0.87 which is reached for 
very small back pressures. But the curve of this 

coefficient of discharge as a function of the ratio -p is 

not at all a straight line. It shows a slight depression 

around t>=0.7. On the contrary, if the ratio of the 

ordinates of the curve for the thin-plate orifice be 
taken to those of the experimental curve CD for con- 
vergent nozzles instead of to the theoretical curve, it 
is found that the ratio increases in a straight line. On 
Plate III points representing this ratio for certain 

values of -p are marked. It will be seen that these 

points be very nearly on a straight line GH, which of 
course is tangent to the thin-plate orifice curve at 
the point A , which is on the ordinate of the point where 
the curve CD becomes horizontal. 

COMPARISON WITH THE RESULTS OF HIRN'S EXPERI- 
MENTS ON AIR. 

In 1885 Hirn made some very precise experiments 
on the flow of air through nozzles and orifices in thin 
plates. It is interesting to compare his results with 
those of our experiments on steam. Hirn used the direct 
method. After passing through a holder the air was 
caused to flow into a receptacle in which a vacuum had 
previously been formed. By a certain method of register- 
ing he measured the fall of the holder at equal intervals 
of time, as well as the initial and discharge pressures. 



44 FLOW OF STEAM 

On analyzing his numerical results, given in the 
Annales de Chimie et de Physique, March 1886, it 
will be noted, first of all, that Hirn did not take 
into account in his calculations the slight contraction 
of the orifice occasioned by the lowering of the tem- 
perature due to the expansion of the air. It is easy 
to make the correction, however. Again, it appears 
to me to be wrong to dry the air after it left the holder 
and before passing the nozzle, because dry air was 
flowing through the nozzle, while the gas-holder meas- 
ured humid air. As the volumetric discharge of gases 
(referred to the initial pressure and temperature) is 
the same for all gases at the same temperature, and 
that of the vapor could not differ very much, it would 
have been better, in my opinion, if he had not dried 
the air. The resulting error would only have been a 
fraction of the proportion of water vapor. What is 
the error to be expected from this cause in Hirn's 
experiments? At 10° C, the mean temperature during 
the experiments, the vapor pressure of water is 0.0125 
kilogramme per square centimeter, and its density is 
equal to 0.00195. The error due to this cause would 
increase the results by about 2 per cent. Hirn's ex- 
periments were made on two convergent nozzles, one 
having 9° and the other 13° convergence; also on a 
conico-cylindrical nozzle, and on two thin-plate orifices. 
The experiments on the nozzles clearly showed that 
the discharge becomes a maximum when the back pres- 
sure falls slightly below the theoretical figure for per- 
manent gases (0.526P). Taking the discharge per 



THROUGH CONVERGENT NOZZLES, ETC. 45 

unit area of the mouth of the orifice, we have the fol- 
lowing figures: 19.80 litres per square centimetre per 
second for the 9° nozzle at the initial temperature of 
15.75° C. ; 19.77 litres per square centimetre per second 
for the 13° nozzle, at the temperature of 8° C. ; 18.85 litres 
per square centimetre for the conico-cylindrical nozzle 
at the temperature of 6.5° C. The theoretical maximum 
discharge for these nozzles, calculated by the formula 
Q = 1.164!T, where T is the absolute initial temperature 
and Q is expressed in litres per square centimetre per 
second, is respectively 19.77, 19.50, and 19.47 litres. 

For the conico-cylindrical nozzle, then, the experi- 
mental discharge is less than the theoretical, which is 
to be expected because of the losses by friction in the 
cylindrical part. But for the convergent nozzles the 
experimental discharge is larger than the theoretical 
by about 0.75 per cent on the average. Here we have 
the same results that we have already observed for 
the vapor of water. It should be noted always that this 
exaggeration of the measured discharge could arise, as I 
have explained above, from Hirn's having dried the air. 

I have plotted on Plate IV the experimental points, 
taking the ratio of the pressures as abscissae and the 
ratio of the observed discharge to the maximum dis- 
charge as ordinates. I have also plotted a few theo- 
retical points. 

It will be seen that for the nozzle having a converg- 
ence of 13° and for the conico-cylindrical nozzle the 
experimental curve AB is very close to the theoretical 
curve except in the neighborhood of the maximum 



46 FLOW OF STEAM 

discharge. But for the nozzle with 9° convergence 
the experimental curve CD departs considerably from 
the theoretical curve, particularly towards the origin. 
It would seem that there was some systematic error 
in the series of measurements relative to this nozzle. 

The results obtained by the two thin-plate orifices lie 
very nicely along the curve EF on Plate IV. Taking 
the ratio between the ordinates of this curve and those 
of the curve AB for convergent nozzles, we find figures 
which increase along a straight line in the same way as 
was obtained in the case of steam. That is to say, the 
points representing these ratios which are plotted on 
the chart lie very exactly on a straight line GH tangent 
to the curve EF for the thin-plate orifice. The origin 
of this line is 0.629, while Hirn found by direct experi- 
ment that for small differences of pressures the coeffi- 
cient of discharge for thin-plate orifices was 0.633. 

Comparing the curves thus obtained for the flow of 
air with those obtained for the flow of steam, it will be 
noticed that they are entirely analogous and can be 
almost superposed. There is always this difference, that 
the theoretical maximum of the curves for air occurs 

at -£ = 0.526, while for steam it occurs at -p = 0.58. It 

will be noticed also that the straight line which repre- 
sents the ratio of the coefficient of discharge for thin- 
plate orifices to that for convergent nozzles for steam 
lies about 2 per cent above the corresponding line for air. 
I am unable to decide whether this difference is due to ex- 
perimental errors or to the different nature of the fluids. 



THROUGH CONVERGENT NOZZLES, ETC. 



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THROUGH CONVERGENT NOZZLES, ETC. 



5i 



fi 














s> 




-ic 




CO 




OJ 


^ 


05 


Ss 


O 


^ 


" 


80 


IS 


O 


b- 

1— 1 


IC 


co 


CO 


TtH 


l> 




i- 




<^ 


O 


<u 









(5 


O 


s 


^H 


oq 


a 




a 


C> 


IK 





~ 


1- 




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? 


£ 


s 


s 


a 





& 


<N 


R. Tt< 


g 


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-to 


1 




^ 







cd 




05 




oc 




T— 








IC 




r 




e 




E 




►0 




» 




t 




i 









,_, 


CO 


CO 


CO 


OS 


Q 




c 


l-H 





c 


l-H 


00 


l-H 




•■ifeift, a 


to 


id 


to 


to 


-t 


to 




K 
















. 


lO 


^ 


Jt> 


,_, 


CO 


00 






00 


00 


00 


OS 


OS 


00 


OS 


b- 


00 


•~! 


CM 


t^ 


c3 




b- 


CO 


CO 


TP 


Tt< 


TH 




P 


x w 


1-1 


<M 


<M 


CM 


CM 


CM 




00 


b- 


00 


»o 


CM 


00 


S 

03 


^3 a) 

« 02 


S3 


•<* 
OS 


os 





to 

1— 1 


l-H 


02 


H M 


00 


O 





- 


l-H 








CO 


CM 





CM 


,_, 


»o 




3© ■ ©f> 


<N 


to 





O 


i-H 


to 


E 


CO 


CO 


CO 


CO 
CO 


CO 
CO 


CO 
CO 




-i 


CO 


to 





00 


b- 


,_, 




°fc«r^ 


CO 


1—1 


to 





-* 


CO 




8&|l «> 


»o 








,H 


l-H 


l-H 






CM 


CM 


CM 


CM 


CM 




H 





















CM 


t^ 


to 


CO 


00 




c 


b- 


CO 


OS 


to 


OS 





2 




H 


tO 


to 


CO 


CO 


b- 


3 


CM 


CM 


<N 


CM 


CM 


CM 


1 


hJ 

















M 














E 


C 


b- 


b-. 


b- 


b- 


OS 


b- 


3 


©-S O 


CO 


rt< 


Tf 


Tf 


t* 


■^ 




c 


cd 


td 


to 


to 


to 


to 




H 
















, 03 


<N 


CM 


OS 


to 





CO 






to 


£3 


T— 1 

to 



to 


OS 



to 






<M 


os 


to 


to 


CO 


CM 




O 


l-H 


00 


00 


b- 


^ 


b- 




"■§ ©-la, « 


^t 1 
lO 





<* 


Os 


to 


co 




O 



















N 














? 


~i ^ 


T* 


CO 


CO 


CO 


b- 


b- 


3 


">* 


■HH 


i-H 


OS 


00 


i-H 


.s«.> M 


CO 


CO 


t> 


b- 


00 


b- 


Ph 


S M 





O 


O 











l^l « 


OS 


00 


os 





CO 


-* 


^3 


l-H 


to 


to 


CO 


CO 


CO 


TO 


3 a 














< 


- ^ 
















6 rH 


lO 


CM 


CO 


■^ 


to 


b- 




» 


lO 


CO 


CO 


CO 





co 



52 



FLOW OF STEAM 







<«. 








>-, 




O 




03 








CI 




l"»«a 




f- 




s 








-^ 




bn 




Cr 




►fid 




cr 




X 


s 






o 




II 


Cn 




'•£ 


I 


X 

03 


Oi 


r ~: 


IN 


re 


l^ 




S» 


r • 










S» 


o 


S 


O 


CI 


<N 


*, 


i— i 


5 

cc 


"8 




e 




ai 


o 


~ 
^ 


o 




— 




CM 


• 




fc 


3 


g 


ri 


CN 




d 

IN 


fc 




s 


g 


w 


« 


■$ 





3 




C 




cd 




Cs 




X 









ICCO!(NCIOOOOOHO 


N(NMNM(MiMN(Nh 


T3 

e? 

o 

s 

s 


Per 

sq. cm., 

W 

gr/sec 

11 


N-trr^rNiNtoooeo 


HHOOH(N(NlNNN 


Total, 
gr/sec 

10 


COOCCCONfN^^OOO 

©NO0CNH00 05O10 


00000000X00 


Total 
Water 
Dis- 
charged, 
kgs/sec 

9 


IOWOOCXOCNHCN 


eocccoeoeoeocococoeo 


Rise of 
Tempera- 
ture, 
h — to 

8 


MCOrwiONiCOCOOWM 
NNiONOOCOOCOCO 

050HNN050CNN 


•s. 

s 


Leaving, 

7 


CiW>O(NO00M00^Tti 


eOCOCOCO<N<M<M<N<N<M 


Entering, 

to 

6 


(Mi-iOOi-Hi-hOi-h^i-h 


;0OOOOOOOOO 


Head, 

H 
Milli- 
metres. 

5 


CO O O CD <N <N 

N0CO5iOHCi(NCTf*C0 

r^^x^tr^oxooo 


o 


OOr-ICOt>OXXOC5 

OHNNMCONOiTi'M 

TtiCOCOC^T-H<NCO'fOO 

c'oooddoood 


in 

s 

Ph 

0) 
j3 
•O 

DQ 




o^ocoxxxxxx 

00iO(MCi(Ml>O5T-(i-iec 

,_j ,-J HOOC C r-1 »H ,-i 


■a "s 

a a 


^r^COCO<M(N<N(M<N(N 




6 ,-i 

ft 


HHH(M(N(N(NlNiMO 



THROUGH CONVERGENT NOZZLES, ETC. 



53 









to O 

00 . 

OS i>, 

© °* 

II 00 



r- 




(N 




t^ 


O 




°o 


(U 


»\ 


fe 


T~\ 


1 


"8 


cq 


c 




S*. 




^ 


O 




o 




co 

OS 




5-. 
3 


i— i 


g 


O 




& 









O 


1— 1 


00 


00 


CO 




o 




•* 


CO 


00 


CO 


t^ 




"■S^ift" 


<N 


CN 


<M 




r _J 


,_; 




































. 




iO 


O 


,_, 


1—1 


oa 




^a s 

Ph .^ 




t^ 




*a 


CN 


00 


^ 


<N 


OS 


iO 


"# 


*"! 


d 




,-H 


o 


iO 


rH 


to 


o 

(0 

s 




tO 


to 


TfH 


■^ 


CO 






tO 


CN 


00 


to 


1> 


a 


3^ 




co 


Tfi 


CO 


i> 


<M 
















03 


o> 




CO 


cn 


to 


tH 


(N 


c£ 


H M 




CO 


CO 


tH 


"^ 


- 




3 03 i 1} 03 
H^ M jS M 




CO 


IO 


tH 


I> 


00 






oo 


CO 


CO 


OS 


CO 




OS 


o 


OS 


OS 


00 


o 


t 




CO 


(M 


CN 


CN 


CO 








CO 


CO 


tH 


to 


CO 




o 03 m o 






CN 


00 


^ 


xh 




Q3 9-bT 


















CO 


*# 


to 


1—1 


T— 1 


CO 






CO 


CO 


CO 


CO 


<N 








CO 


oo 


oo 


CN 


to 




fi 




o 


o 


CO 


CO 


<M 


73 

a; 


03 


1> 


o 


,_H 


b- 


t^ 


OS 






^H 


"* 


co 


CO 


<M 


03 


h^ 




























a 


bo 














a 


cs . 




b- 


CN 


^H 


IS 


<N 


£ 


p o 


o 


00 


00 


00 


00 


00 




C 




iO 


tO 


to 


to 


id 




H 
















_r i °5 




00 


CO 


l> 


1> 






w s a 




<m 


OS 


oo 


t^ 


CN 






rJH 


CO 


CO 


CO 


^ 








IO 


to 


o 


to 


^ 




o 




00 


to 


<N 


l> 


T« 




Ph 


^ 


(M 


OS 


OS 


CO 


CO 




CO 


CO 


-* 


to 


to 






o 


o 


o 


o 


o 


CO 






LO 


IO 


to 


to 


to 


3 




IO 


CO 


00 


^H 


C5 


CO 


CO 


CO 


00 


o 


to 


Ph 


jrH 03 

Ph M 

ri4 




;^ 


? 


* 


oq 


- 


0) 


6 




tO 


iO 


to 


to 


to 


3 




CO 


CO 


CO 


o 


00 


o 

to 


<M 


1— 1 


Tt< 


00 
CO 


00 
CO 


CI 
<N 


OS 


o 


,_, 


<N 


co 




fe 


CN 


CO 


CO 


CO 


CO 








» H 


1—1 


rH 


7-1 


1-1 



54 



FLOW OF STEAM 



^ 



00 § 

O Cj 

© ^ 

II °° 



o 



2 g 



S OS 







CM 


_ 





"t 


to 


CM 





CO 




O 


tO 


CO 


CO 


Tfi 


to 


t^ 


•0 


to 






^ 


"* 


id 


tP 


■«* 


«tf 


^f 


M* 




1—1 


















T3 


. 


CO 


.« 


Q 





c 


1— 1 


00 


CO 




CM 


O. 


oa 


oc 


Ttl 


t^ 


OS 


t^ 


B 


£°£\ - 


lO 


00 


w . 


00 


■0 


CO 


CS 


•"• 


e3 
A 

s 


c* U 


O 





CO 


J> 


">* 


c 


0. 


•0 


so 


I> 


to 


* 


t^. 


50 




t^ 


CO 


G 




CO 





TF 


J> 


00 


CM 


00 


1> 


£ 







to 


CO 


CO 


to 


*Q 


OS 


10 




-~ «D ' O 


















o 


o\ -* 


OS 


CO 


OS 


I— 1 


l> 


T— | 


"tf 


00 


OQ 


H& 


CO 




t^ 


Tf 


1— 1 


OS 








— ?» 


cb 


,_, 


CM 


CM 


CO 


^_ 


00 


OS 


"35 . eg 


T— t 


CM 


X 


"# 


T— 1. 


c 


OS 


§ 




CO 


® 


»c 


O 


CO 


CO 


10 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 




<— u 


•O 


00 


00 





00 


CO 


l> 


CM 






00 


CO 


T-< 


1— 1 


OS 


CO 


c 


tt 




1 = 3 1 00 


-* 


00' 


Tf 


id 


©' 





CO 


^ - 




g£*~ 


CM 


1— 1 


rH 


CM 


CM 


1—1 


CM 


CM 




M 


1> 





10 


CO 


10 





1^ 


^j, 




^ 


OS 





CO 


i-H 


OS 


CO 


OS 


CO 


DO 


S~ *■ 


O 


*# 





1-H 


CO 


CM 


,_j 


t^ 


3 
1 


GO 


CO 


CM 


<M 


CO 


CM 


CM 


CO 


CM 


o 


t& 


















g 


S 


CM 


CM 


t^ 


r^ 


t^ 


i> 





Ol 


£ 


£,? «3 


'"I 


1-1 


•""J 


OS 


OS 


OS 


OS 


os 


"£ 


CO 


CO 


CO 


tO 


10 


tO 


to 


tO 




w 






















CM 




iO 


10 




CM 


t» 






5"-, — is 10 


lO 


tO 


i—i 


CO 


CO 


00 


OS 


os 




fl^SI 


00 


00 


I> 


OS 


0T; 


J> 


t^ 


- 




"3 


tO 


■O. 


tO 


10 


to 


to 


to 






,_, 


00 


t^ 


CM 


c 


"* 





X 







O 


"tf 


00 


O 


t^ 


os 


CM 


1> 




'1 alft, -* 


O 


OS 

0. 





S 








CO 




CO 

c 






O 

























-a 1 


OS 


OS 


OS 


OS 


os 


OS 


OS 


OS 




CO 


CO 


CM 


TP 





CO 


!> 





5; 


flSVS. cc 


CO 


CO 


CM 


CM 


CM 


i-t 


1—1 


H 






















£ 


h j? 





c 


O 


O 





O 








Pn 























3 1 





CO 


CM 


iO 


"* 


CO 


10 


<* 


3 


co 


OS 




oa 


"* 


CM 


OS 


1> 




m 
< 


•J & 


CM 

to 


00 

CO 


OS 
CM 


CO 

10 


TT 

"# 


CO 


id 






6 r-i 


"* 


tO 


CO 


l> 


00 


OS 





_, 




2 


s 


"^*. 


T 


■* 


TT 


■* 


LT 


tO 



THROUGH CONVERGENT NOZZLES, ETC. 



55 



?*> 






OS g 






O ^ 



|3 

•s fe 



1 




o 


00 


00 


o 




o 




CO 


>o 


CO 




.-._.= „ 


X 




tf3 


CO 




03— ^ - 1 












P4 '■ 


o 


o" 


° 


d 




i -J a B 


CO 


1> 


OS 


IO 




$2 -8 


"* 


to 


oq 


OS 
















O 


t^ 


OS 


<M 


-6 


io 


"O 


to 


t^ 












c3 

Si 
o 

S 

0> 


. 


<N 


IO 


OS 


CO 


cr bo 

02 


CO 
00 


1> 

cd 


o 

CO 




TP 


CO 


CO 


<N 












w. 




tH 


-tf 


OS 


CO 




^3 « 

S GO © 


00 


OS 

CO 


d 


1> 




'•:Bm 


00 


CO 


CO 


^ 






OS 


o 


(N 


"^ 




t^ 


OS 


<N 








CO 


CO 




1> 




8 g I L °° 


o 


o 


IO 


CO 




OS 

1> 


1— ( 
CO 


1> 


o 

00 




£ <» 03 ~ 


1—1 










bO 


o 


o 


IO 


o 


m 


S3 


»o 


J> 


00 




3 
* 


'>•£• t^ 










S3 


CO 


oo 


<N 


CO 


J 


<M 




<N 


cq 




o 


o 


T-( 


1> 


§ 


'J~ ° 


CO 


CO 


T* 


o 


H 




"3 


IO 


id 


id 


Head, 

H 
Milli- 
metres. 

5 


1— 1 


<N 


o 


tH 


iO 


00 


<N 


CO 


<tf 


TfH 


<N 


tH 




. 


"<* 


, 


o 


iO 




o 


CO 


OS 


CN 


CO 




"■§ ain. « 


I> 


00 


OS 


OS 




ti 


o 


d ' 


d 


d 




■a * 


CO 


<N 


CO 


CO 


CO 


OS 


T— 1 


<N 


CR\ « 


OS 


T^ 


t^ 


00 


05 


JH ai 










Ph . 

-*> 


4d 


<N 


CO 


CO 


TfH 


3„ § 


TtH 


<N 


•TJH 


o 


3 


oo 


OS 


o 


o 


1 


-3a,\ <n 










la 


CO 


CO 


^ 


id 


<J 


•^ -* 












6 


CM 


CO 


TtH 


iO 




fc 


rH 


'- 1 


i-H 


rH 



56 



PLOW OF STEAM 



e 




c 




O 




tSi 




-^ 




c© 




OS 




OS 




o 


§ 


II 


« 


s 


BE 


g 


OS 




1— 1 


10 


oc 


PC 


I— 1 


r^ 




5- 


O 


«3 


o 

O 


Ci 




•~ 




M 




tt| 


a 




a 




w 




^ 


o ^ 


o 




i— 1 


g 




£ 




OS 


3 


^H 


e 


kC 



R. h 





6s 


.-1 


q 


--c 








6c 




w 






CO 




OS 


o 


oc 


S 



| 


Ratio, 
W 
Wm 

13 


N M N •* IC t)- iff N ffi 
l>Tt^t>CCiCiCC0CCiO 

ooocooooo 


o 

5 


Theoretic 

Maximum, 

Wm 

gr/sec 

12 


©050CNIOHN05 
OS00COiC00^^iOl> 


HOCOCOOCMCCN 

oi>ocoooccic;o:oo 


Per 

sq. cm., 

W 

gr/sec 

11 


(NOOlNNOiCNC'Cl 

HOOCOOCOCONCOCC 


Total 
gr/sec 

10 


©HCCOMNMNCO 

00^0<NW*h(N 

(N(N(MOOHOffi>CiO 
OOOt-hCSOOCSIOiOCS 

1—1 


Total 
Water 

Dis- 
charged, 
kgs/sec 

9 


)t^»CCOi-HTtHTHl-ll-ITj< 

O^OO^iOiOO^iO 
(NCOOOhOWOOJ 

coeoiNcoeococci-KM 


Rise of 
Temper- 
ature, 

8 


OOOOOOQOhhwh 
iO O CO 00 CC (N N 00 Ol 

CDNrt<OtD03HHO 

HH(N(NHHHIN(N 


CO 
g 

m 

a 

a 


Leaving, 

7 


ON00OO00MO»0 
THNOiiOCONCOO't 

(NHM(NCvl(NHiN(N 


Entering, 
to 

6 




Head, 

H 
Milli- 
metres. 

5 


00 00 03COCO»OiOOO 
lOOiffiHTtHOON© 


6 

03 


(NMONCO'tO'OOO 
iC>0f0 05(N(NNNO 
OOC50000C5CSC5C5C5 

ddooooodd 


S3 

02 

to 

j3 

"o 

03 


s a 


COrtii-HCO(NCOCDCOCO 

OffloOHcocwos© 

COHiOHHNININ'* 


COiO^iOkOtOCcOiC 


Initial, 

kgs/cm 2 

2 


cocq<Nioco^cocoio 






Ot^OOOSOi-KNCOrtH 

HHHH(NN(NIN(N 



THROUGH CONVERGENT NOZZLES, ETC. 



57 








s 


OJ 




O 


5> 




oo 




CO 


5 


T— 1 


£ 


CO 


T^ 


^ 


CO 




r^ 






o 


w 
g 


h 


£ 


rH 


cq 


S 




a 




v 




*. 


o 


^ 


o 




CO 




H 

E 


s 

5 




n 




<M 


*3 





a 



« 





PS U 


OOt^lO^TtHfMrHCOfNfNr-I^HCOOOCC 


OOOOOOOOOOOOOOO 


9 

A 
o 

m 

s 

1 

03 


Theoretic 
Maximum, 

gr/vsec 

12 


COt>iOOCOtO<Ml>tOOiOOCOi-iOOO 
COcOcO(N'*COOiO'-HCOCO(M(MI>Oi 


01>»0(MOiCOOCOOi05(MC^05CC05 


Per 

sq. cm., 

W 

gr/sec 

11 


rHTtlOOCOOOCOCOI>050500'^»-HOCD 

•oocc^cocooeoioiMcooieoosoo 


>O^COOOiCOOM"3^0HO®OM 

CCMNHtqHHHHNHHHHH 


Total, 
gr/sec 

10 


NOMCONcOCOOOOON^MtOCO 

ouNooNNHcoaiMcoai'OHffio 


(NNHiONcOOONMMO©H^ 
OiO<MOOr-ioOCCH>COC5iO*OOOC5CO 

T-H rH i-l rH 


' 
F 


Water 

Dis- 
charged, 

kgs/sec 

9 


©NHN©OOONNOCOHHCO 

NN^CON^OlNiO^eOlNTtOO 
OONOHOOOHIMNOOCOHINM 


<N(N(NCCi(NCOCOCOCOCOrHrHCOCOeO 


Rise of 
Temper- 
ature, 

8 


ioo^co-*i>ioa)Tticob-cooocoio 

Tt*»OiO^(M(M(MOOCOOOO^O(N(N 


OOc0t^c000(NC0C00i05rH0000<M 

COCONHNHHHHHrtNHHH 


u 

u 

a 

1 


c 


NNHOCOONHCOOO^IOOION 
05 J> HO NN00(NOHHC0^»O>O 


rHrH(NC0rHC01>Oi00lO^COC0COI> 

^^C0(NC0MHHHIMNIM(N(NH 


Entering, 
G 


(NNNNIN(NN(N(N(NN(N(NN(N 

iOrH»-0»0»0*OiOCOCOCO©C5COeOCO 


»OiOiCiO^O»0»OiO»O^OiO"^iO>0»C 


Head, 

H 
Milli- 
metres. 

5 


lO iO to Jh- lO CO 

MOONNO^MOON^H^ON'* 

l^TtlCXlCOt^r-llOt^t^r-liOO^lOOO 
COCOCO^CO^^^^TtlrHrHT^TtlTtl 


c 


^MHOOH(N(NCO^O(NMH(N(M 
t*iOO-^iOOOC1CCOOOOOOGC'OOCO 
!>00OiO5O5O5O5O5OiO5O5C5i00J>00 


OOOOOOOOOOOOOOO 


g 

3 
m 

Ph 


53 ~ « «« 


C0i0^rH<NC000O^O<Ni>l>c000 

^(NiOMOiOiQiCffliOlNOOHiroCO 
0)ONNNOl^X©©0)fflNlNH 


H(NN(NN^iO»OCO^^Tt<HHH 


Initial, 
kgs/cm 2 

2 


NW(NiOiOiOOiOCO»CON«OON 

NNOOOOOiOCiONOOffiiOM 


(NCOMiM^iOiOIN^Tti^iOHHH 




O ,H 


050HNM^IOC000050HNCC* 



58 



FLOW OF STEAM 



CO 
03 


s 


08 


'- 


c 


o 


II 


•-C 




co 


S 


CO 


*Q 


"tf 


re 




t^ 






ta 


u 

g 


o 


K 




oq 


S 









w 


O 


5. 


o 




r^ 








^M 


s 


3 


s 


♦a 


q 


*. 


?* 


| 











S 





N 2 


CO 1 i - r- X C C -f* rH (N rt CO H lO O C CO 

OG©XCOa©WcO'<tCOM(NCNHrt 

o o o o c o c ©-© odoocood 


§ 

s 
3 

a 

o 

02 


Theoretic 
Maximum, 

gr/sec 

12 


't00OOHO'*O«'fl^rHCO'*HTtfH 

CONNOOCMOOCNCNQOONMNO 


OOOOONNTHiOtN^NCNOJONiOcO 
i-ti-ii-!i-ii-iM<N<NCNICOOOOOOOC5C5CiCl 


Per 
sq. cm., 

gr/seo 

11 


T^iO^TfiOC^OiOt^CSrfH-^eOTtiOOOCO 

CNTfiOrt^MMOOMO'CMi-itCJ 




Total, 

gr/sec 

10 


OOCOlCOcONOHNOOCi^HNNoOH 


HC5HNiO(MiMOONNMONC«!HO 

XNNO>OHH050©OTt<NO«lN»0 
i—i i— i i— i i— i i— i i— i 




Water 

Dis- 
charged, 
kgs/seo 

9 


ffliOCOCOffiNLOHOCNONMMMNO 

HOSNOSOOHO^NCiNNOrHOOrti 
Tt<M^M^(NiMlNHOiCNXffirHCNM 


C0CCCCCCC0C0CCCCCCC0CM<N(N<NC0CCC0 


Rise of 
Temper- 
ature, 

8 


XMLCOONNOOCCLOOMON^M 
OXniOMCiHiNcCMXX^N^HO 


iO'*CO(NOi-i(N05C35C50'*CiCOOOTrii-i 

THrH^-li-Hi-KNCNl-li-Hi-ITtiCCCMfNi-ll-li-i 


tn 

£ 

o 

a 
S 


Leaving, 

7 


OONNM005XOOOON0010005COW 
i0C0OC500CCiOOOl>G000CM0iOC00C 




Entering, 

to 

6 


<NCNl>.l>I>t^(N05CqcM(N<NCqO(MC>JC 


iOiOiCiO»0»OiQiO>OiOiO»CiOiOiO'C'C 


Head, 
H 

Milli- 
metres. 

5 


iO >0 ifl iO O »0 WN LO 

i-IC0r-4f0Jt^t><NCC(Nl-Hl0lCl>05<N^O 

N HNHHNOffl^NOMiCXCOXO 


o 


OCNrH^iOOHN^HXXNMffiMiC 

looofflco-tx oa oa »o c n x x o d 

OM>NXiOOONX050000ffiOi 


ooooooooooooooooo 


go 

i 

m 
m 

2 

Ph 
o 

3 
1 

4 


•jH in 

[=4 M 


NHONMfflNOOTttffiiQiflNC^Cl 
MNiOMOCNMCNOOLOQiflMMM 
NOONOOffiOOMfflfflOOOHMiCO 


OOOOi-HOl— It— li-Hi-HlOOlOOOOO 


Initial, 

P 
kgs/cm 2 

2 


i-i^NOOO^OOOfNcO^iC^COXN 

MNOHHCOLONHfllOHNTiilOO 


HHHHHHHHHWIOOOOOCO 




6 


CCNXOiOOXOiOHtNM^iOCNX 
iCiOiOiflOOC«iht>-r«NNNNl>N 



THROUGH CONVERGENT NOZZLES, ETC. 



59 



o 

cS 



°> » 

O °5 

II «o 

. 00 
s£ CO 

l§ 

<N .. 

CO . 

-IS 

© so 
*g 

O^ 

b os 
i5 "* 






3 

O 



«o 





'5kl S CO 


tCr^cOTHC50ir^05cO^I>l> 
Oi00cD>O(MCir^cO'-H>OTt<cO 
OiC5CSO5CT>00000000r^cO»O 

ododdooododo 


T3 

» 

03 
A 
O 

a 
1 

0) 

0Q 


Theoretic 

Maximum, 

Wm 

gr/sec 

12 


OMnNOWOOHOOH 

TfTlH^tlTtH-^'^TtlTtllOlO»0>0 


Per 
sq. cm., 

W 
gr/sec 

11 


Ttl^COi-lOSGOOOOOTtHrHCO 

OiOH00^®N03N(M00N 
TfiTtH^tiCOC0<M(N(M(MrHOi00 


Total, 

gr/sec 

10 


ONCOOMNONHOOIM 
COOOOONOJiO^OiON 


NcO(MON^ONCD00iO»O 
(NNNNHHHHOOIOON 


1 


Water 

Dis- 
charged, 
kgs/sec 

9 


OiiOGOOiooT-ni>a>i>>oi>oo 

©CONOOHiOcDiOHOOCO 

m>jnnooooooooo500h 


<^<N(N(NC^(N(N(NC^C0C0C0 


Rise of 
Temper- 
ature, 

8 


0»ONNN00^05HNH(N 

rH0000r-lT-IQ0COI>OiC0T-ICX) 

OOi00001>iOiOiOCOT-iOOiO 
CO(N<N<N<NCSI<N<N<N<Ni-ii-i 


to 
0> 

a 

a 


M 

G 

3 


0»OkOiOiOcO(MN03i0050 
ONNOONM(ON(NOiN 

COCOCOCOCOCOCOCO<NO*<N<N 


Entering, 
6 


oooooooooooooooooooooo 

OO00000000G00000C00000 


Head, 

H 
Milli- 
metres. 

5 


CO CO CO i>*o 

^TdHcOCiCO^OOcOOCOCOCi 
COCOCOCOCOCOCOCOCO^^^ 




OcOiOCOOOOOcOCOcOOi-H 
ON»COOr- i CO CO CO O ^ C5 <N 
COCOcOcOl>l>l>t^000000C5 

o'doddodooddd 


10 

e 

(0 

in 
Ah 

J3 

'o 

02 


IN 

•i a 

03 O 

G &\ CO 

V 1 02 




c0OcDc0i>l>l>l>00G00i05 


3L i 

■Sfti-S. N 
1-1 M 


H03HN -*'(35 Oi TH <M CO 1> 00 

MN(N-hhOh^iO>CON 


oooooooooooo 




i 


lOCOtXXDCiOr-KNCO^LOCO 
OOOOOrH^HrH^-lr-I^H^H 



6o 



FLOW OF STEAM 



3 



00 ^ 

© & 



s 


CT- 


£ 


i— 


03 


rc 


C) 




1> 




& o 






(U 


o 


s 

o 


o 

1—1 


CI 


•u 


c^ 


e 




e 




w 




n. 


U 


^ 


o 




rH 




CM 


£ 


t 


? 


3 






~J 


g 




<u 


n 


^ 


IN 


s 


fa 




*a 




43 


w 

3 


1 


BO 


•~ 


3 


Q 


o 




CO 




OS 




00 





• 




CNHNTflfiOCHHNO© 

NNNNOCCOI-hMNh 

0CXX0CCC0CXX0CNNO 

cocoooocdddd 


t3 

1 

83 
g 

s 

I 

-S 


Theoretic 

Maximum, 

Wm 

gr/sec 

12 




io*oiciococococoeococoeo 


Per 

sq. cm., 

W 

gr/sec 

11 




hhOCshiNNNNNCO 


Total, 
gr/sec 

10 


NHOOONi-iXOiCiCCO 
CCCOOCOOOCOCQOO 


E 


Water 
Dis- 
charged, 

kgs/sec 

9 


»OW00W50000NHCNiO00 

XOOl'tNcOXHlNXrfH 


coeocococococoeococococo 


Rise of 
Temper- 
ature, 

8 


MCO-h»CNIC«XMCOCCO 


OSOhNNOOONNNW 
NMMMHHMMHrtHrt 


a5 

£ 

© 

G 

s 


Leaving, 

7 


OCOiOINOOCMX^TfON 

iCCNX^kOOCMCOMOi 
COMMCO(N(NMN(NlN(Nr-i 


Entering, 
to 

6 


(NM^(NN(NNW(N(N(N(N 

(Ni-lOCi-HT-iOr-i^^— IrH 


OCtOOOO'OfflOOOC 


Head, 
H 

Milli- 
metres. 

5 


CO 10 LT C (N N (N X 

NXCLOHCNO^COOffi 
N^X-tNCXCC^cOiC 


•J B,|Ah t* 


OOWCONCXX^CC 
MOX'OfNXNCCr-'XON 


ocoooooooooo 


• E 

m 
m 
O 

-2 

J3 

"o 

■a 

< 


f. z 


Ot-hoocooooooooooooooooo 

lO "* CC CB OS CN *Q ~r CO J> O CO 

httNC(NNC:hi-icoCN 

d d d d d d d i-h »-! t-I r-J i-I 


3- § 

'2 M 

1-1 M 


OCOX^'OiflO(NM»ON 
t* -tf CO CO <N <N <N <N <N <N <M* (N 






NXffiOrHMeO^iQONX 



THROUGH CONVERGENT NOZZLES, ETC. 



61 



^ 



-se 




ti 




c> 




i-Ss! 




O 


g 


on 


<o 


Gi 




o 


S3 


II 


00 




rH 


> 


Oi 


g 


I— 1 

co 


r^ 




(M 




h- 


• 


fc. 


o 


ft) 







o 


6 


OJ 


o 




Ik 




« 


s 


cq 


Q 




ft) 




5«. 


o 


^ 


o 




CO 


6 


ft) 


§5 


£ 


<N 









d 


^ 


s~ 




'w 








ft) 


50 
"S3 




go 





S 




o 




to 




Oi 




GO 





o 
'■5u|ft co 


^rt<00©©NTt<OMNHMC^ 

ooooooi>i>i>i>t>cc>*o^Tt<coco<N 


ooooooooooooooo 


T3 

(1) 
M 

e3 
-G 
o 

CO 

s 
i 

CO 
0Q 


Theoretic 
Maximum, 

Wm 
gr/sec 

12 


COcO^O)hoOOMhNcOH(MI>N 




Per 

sq. cm., 

gr/sec 
11 


(MOiiO^r-t^iHrtiO^^iOiOCOO 


Hoio^'O'-t^fficoio^coHOoo 

»OiO^TtHCOCOCOCN»OlC<lCN104eSICNJi-H 


Total, 
gr/sec 

10 


^COOOiONMHOOOOO^COOOMO 
»O^MN(NO)OrHOOOHCNOOO)iO 


COCQ>O^H<MO)C5CO^'-iCX)i000^1> 

©CO^-*HOO®OOlX)NNCtCiO 

T— 1 1— 1 H !— 1 1— 1 T— 1 1— 1 


E- 


Water 
Dis- 
charged , 
kgs/sec 

9 


M«5^NOO!OOiN©CDHOMW^ 
00C0C00iO(M00r^0l01000iO5<M00 

OOCSOOOOO)Ohhhhh(N(N 


MNNNMWNMNCOMMMMM 


Rise of 
Temper- 
ature, 

8 


COCOTtHiOCO0000CO00000000T-ii-i(N 

'-KNoo^Tjic>>coeo05<Mi>'-<Oioeo 


T)HlOrHHCOCOCOaiCDCD10»OOlWH 
COMMCOMNINHHHHHHHH 


H 

2 

ID 

a 

a 

0) 

H 


Leaving, 

7 


M0000N»0OO00»0»COOhhN 
OOtDCO(MHCN|rH©ffli005CDNO 


OHNNOO50)lCcMrHr-lO0500N 

TjHTtC0CO<MC>><N<M(M<NCq(NT-i^H^-( 


Entering, 
to 

6 


NN^N(NNNCNNN(N(NOOO 

00CO000000000O00cOCOt^l>l>l>l> 


I0i0i0>0»0t0>0i0>0i0i0»0i0>0»0 


Head, 
H 

Milli- 
metres. 

5 


COCO iO*O»OiOC0<N 

OOCONNHOOWCOiOOCO^tMO 
MOJONMOONiOiO'OiOiOcOOO 




05CO(NM^IO(NNCOOOOIOCOO»0 

c<icicicocoo^cooocoi>oco>oco 

C0C0^»O»O»OcOcOl>00cX)OiOiO5Oi 


ooooooooooooooo 


en 
g 

m 

la 

"o 
4 


■a " a 

fl SVx. CO 

V rt CQ 

P=< bO 


lOioioioioioioooaHcocoMO)'* 

iCMOO^OlO^N^HMHHOH 

eoooooiocx)OOcoi>o»OT--iTf<oo 


rH-iH T-i(Ni-(T-i(Ni-i(NCMCOCOTtHTtHrhi 


Initial, 
P 

kgs/cm 2 

2 


I0>0t0i0i0i0t0i0i0»0t0ici0i0»0 

cococoooocooocoooco oo'oo oo co oo 

HH0000®OHCD05^^00C0cOOi 


^^COCOCNCOCOCNCNCOCOCO-^^'* 




1 


OlOHMeO^iOcONOOOiCHIMM 
(MCOCOCOCOCOCOCOCOCOCOTti-^rt<r^ 



NOTE ON THE FLOW OF HOT WATER 
THROUGH NOZZLES. 

In this short note I propose to analyze the 
phenomenon of the flow of hot water through a con- 
vergent nozzle and to explain the results obtained by 
Sauvage & Pulin in experiments made by them in 1892/ 
which appear very singular at the first glance. 

The theory developed in the preceding work relating 
to the flow of steam can be extended to the somewhat 
more complex problem of hot water which is partially 
vaporized during the discharge. The most simple case, 
which I will discuss first, is that in which the water in 
the high-pressure receptacle is initially at exactly the 
temperature T corresponding to the pressure P of 
the steam; that is to say, just on the point of evapo- 
rating. It will then begin to evaporate as soon as its 
pressure and temperature fall. A remarkable circum- 
stance gives to this phenomenon a very special interest, 
namely, that the quantity vaporized on the one hand 
and the velocity of flow of the mixture of water and 
steam formed on the other hand are both practically 

1 Ed. Sauvage, Ecoulement de l'eau des Chaudieres (Annales 
des Mines, 9th Series, Vol. II, page 192). 

62 



FLOW OF HOT WATER THROUGH NOZZLES. 63 

proportional to the fall of temperature. Considering 
the entropy diagram/ Fig. 3, let AD be the curve of 
the entropy of water, and EF that of saturated steam. 
Let the point A represent the state of the hot water 



To 




T1 1, 




Fig. 3. 



at the temperature T and the corresponding pressure 

p.. 

During the expansion in the nozzle a small quantity 
of steam is generated which forms a more or less homo- 
geneous mixture with the remaining water. It may be 
admitted that the expansion is adiabatic on account 
of the very short time during which the fluid remains 
in contact with the walls of the nozzle. This adiabatic 
expansion is represented on the entropy diagram by 
the line AG parallel to the axis of temperatures. When 
the mixture has reached the temperature T t differing 
from T by the quantity 6, the proportion of steam 

1 The utilization of the entropy diagram for the vapor of 
water was made the subject of some remarkable articles by 
Prof. Boulvin in the Revue de Mecanique in 1897 and 1901. 



64 FLOW OF HOT WATER THROUGH NOZZLES. 

generated is given by the ratio of the segment of the 
straight line DG to DF. 1 Now the curve AD of the 
entropy of water is practically the same as its tangent 
at the point A, if the fall of temperature is not large. 
DG is then proportional to the fall in temperature 6, 
with a tendency, however, to increase a little quicker 
than 0, and one can write that the ratio of DG to DF 
is very nearly proportional to d; that is to say, 

x = ad, 

a being a coefficient which depends on the initial tem- 
perature T . The value of this coefficient can be easily 
ascertained if the entropy diagram be drawn to a 
large scale. 

On the other hand the velocity of flow V at the 
moment when the temperature has fallen to T x is 
given by the area of the triangle AGD. The base of 

T 

this triangle is approximately equal to Xjp- } r being the 

■* 

heat of vaporization of the water. We have then the 
following relation: 



whence 



1 



V^0-JgEa w . (1) 



1 We may safely assume that there is no retardation of the 
vaporization so that the quantity of steam in the mixture 
is that corresponding to the segment DG at each instant. 



FLOW OF HOT WATER THROUGH NOZZLES. 65 

We will now calculate the area of nozzle necessary to 

i allow unit mass of wa ter to flow when the temperature 

has fallen by 6°. This area is equal to the specific 

volume v of the mixture of water and steam divided 

by the velocity V. 

x 
The specific volume v is equal to 1 — £+77, D being the 

density of saturated steam at the temperature T t = 
T —d, or replacing x by its value ad, we have 

v~l-aO+jf (2) 

Dividing v by V, we have the area S: 






For any given initial temperature a and r are fixed. 
In the expression for S, consequently, the only vari- 
ables are the fall of temperature 6 and the density D 
which is a function of T —6; consequently £ increases 

or decreases in proportion to the quantity ^+ ^p and 
since a is very small, S is practically proportional to 
this quantity. Now when 6 increases -n decreases. But 
on the other hand the density D diminishes and con- 
sequently the second term jt increases, so that a time 
arrives when the increase of the second term compen- 



66 FLOW OF HOT WATER THROUGH NOZZLES. 

sates for the decrease of the first, and the quantity 

la. _ " 

7f+-R then passes through a minimum. 

In order to ascertain the position of the minimum it 
is necessary to express D as a function of the tem- 
perature. 

In default of a simple relation between these quan- 
tities the problem can be solved in any particular case 
by the aid of Regnault's tables; for example: 

Suppose the initial pressure P equals 10 kilogrammes 
per square centimetre. The corresponding temperature 
T equals 178.886 + 273°. By the entropy diagram the 
value of a is found to be 0.00216; the density is given 
by the tables. It should be noted that D must be 
expressed in kilogrammes per cubic decimetre. 

The following are the values of the quantity -5+ jr 

for increasing values of 6 in the neighborhood of the 
minimum: 

For 6 



6°C. 


-^+^ = 0.6501 


7° 


0.6374 


8° 


0.6300 


9° 


0.6274 


10° 


0.6293 


11° 


0.6325 


12° 


0.6355 



Plotting the curve, it is seen that the minimum occurs 
for = 9.2°. The absolute pressure corresponding, to 
this value of is S.044 kilogrammes per square centi- 



FLOW OF HOT WATER THROUGH NOZZLES. 67 

metre. Consequently when the pressure falls below 
about 80 per cent of the initial pressure it is necessary 
for the nozzle to diverge in order that the mixture of 
water and steam should continue to expand. 

If the nozzle is simply convergent, the pressure at 
the mouth of the nozzle will be exactly equal to the 
value just obtained (i.e., 8.044 kg.). If the discharge 
takes place into a space in which the pressure is much 
less than this, as, for example, into the atmosphere, a 
sudden expansion occurs which is much stronger than 
in the case of the discharge of steam alone. The jet 
is seen to swell suddenly and take the form of a very 
wide paraboloid, as is well shown by the photographs 
reproduced in M. Sauvage's memoir. 

The lateral expansion of the jet is enormous at the 
mouth of the nozzle because, as we shall see, the speed 
of flow of the mixture is not high and it is proportion- 
ately greater as the pressure in the exhaust space is 
lower than the pressure p giving the maximum output. 
The discharge per unit of area of the mouth of the 
nozzle can be calculated from the pressure p which 

makes the quantity -n + j^ a minimum. Making this 

calculation for the case under consideration, we find 
first that the quantity of steam generated x is equal to 
1.99 per cent; that the specific volume of the mixture 
reaches the value 5.78; that is to say, at the mouth 
of the nozzle the steam occupies a volume 4.78 times 
that of the water ; that the speed of the mixture at the 
mouth is 28.55 metres per second; and finally that the 



68 FLOW OF HOT WATER THROUGH NOZZLES. 

weight of fluid discharged is 494 grammes per square 
centimetre of orifice per second. 

It is interesting to compare this last figure with 
that obtained if the nozzle was discharging initially 
saturated steam. In that case the quantity discharged 
would be 143 grammes per square centimetre per 
second. In the case of the hot water, therefore, the 
weight discharged is 3.5 times larger. 

If the expansion is to be continued in the nozzle, 
the throat must be followed by a divergent portion. 
In Fig. 4 the curve CMD of the areas as ordinates is 



5CM 




5° W 

FALL OF TEMPERATURE t 4- 

Fig. 4. 



given as a function of the fall of temperature for the 
case where P is 6 kilogrammes per square centimetre, 
the water being initially at the temperature of vapor- 
ization at this pressure. 



FLOW OF HOT WATER THROUGH NOZZLES. 69 

In the case of the flow of initially saturated steam, 

it was found that the value of the ratio -p corresponding 

to the minimum area (that is to say also, to the maxi- 
mum discharge) is equal to 0.58, and that it is sensibly 
independent of the initial pressure P . In the case of 
hot water we have found, when P equals 10 kilogrammes 
per square centimetre, that the value of the ratio cor- 
responding to the minimum area is equal to 0.804, a 
value very much larger than the preceding. On repeat- 
ing the same calculations for different initial pressures 

v 
it is found that the value of the ratio -p corresponding 

to the minimum varies considerably. It approaches 
unity as the pressure P decreases; thus for P = 6 
kilogrammes per square centimetre we find that the 
fall of temperature 6, which gives the minimum 

value for #— 77, is 7.15 degrees, whence -p = 0.829 ; the 

speed of flow at the mouth of the nozzle is now only 
22.5 metres per second, and the discharge 344 grammes 
per second. 

Let it be supposed now that the water is initially at a 
temperature lower by t° than T , the temperature of 
vaporization at the pressure P . As long as the fall 
of pressure is insufficient to cause evaporation the 
water will remain in the liquid state and the discharge 
will follow the laws for liquids. When the pressure 
falls below this limit, however, partial evaporation 
will take place, and again a point will be reached at 



70 FLOW OF HOT WATER THROUGH NOZZLES. 

which the increase of the specific volume will com- 
pensate the increase of velocity. At this moment the 
area of the flow will become a minimum. The decre- 
ment of temperature t has the effect, then, of rapidly 
increasing the possible discharge per unit section of a 
given nozzle. We shall see later, by an example, that, 
except for very small values of t, the minimum area 
corresponds exactly to the point where evaporation 
begins to occur. 

Calling P f the pressure existing at the point where 
evaporation begins (namely, the pressure correspond- 
ing to the temperature T —t for saturated steam) and 
6 the fall of temperature from this point, the speed of 
flow is given by the relation 

V 2 P — P' 1 r 

r-V + 2^ ! iw' • • • < 4 > 

D being the density of the liquid and a having the 
signification already given. The specific volume of the 
mixture is always given by the expression 

7-l-ofl+g. 

The analytical discussion in this case is even more 
difficult than in the preceding; but any particular case 
can be solved by means of Regnault's tables. It will be 
seen from the above that only a slight fall of tempera- 
ture is sufficient to increase the discharge enormously. 
Thus, for example, to obtain a discharge of 494 grammes 
per second for a pressure of 10 kilogrammes per square 



FLOW OF HOT WATER THROUGFI NOZZLES. 71 

centimetre, the speed must increase to 4.94 metres per 
second. For this speed 

P -P' = 0.124, whence Z = 0.54°. 

This low value clearly shows the considerable influence 
which the decrement of temperature has on the discharge 
from a convergent nozzle. 

Now in different parts of a boiler the temperature of 
the water is always a little lower than the temperature 
of vaporization. This difference will easily amount 
to several degrees, and is very variable, not only 
from point to point in the boiler, but also from one 
moment to another, depending on the state of the 
internal currents and the activity of the evaporation. 
One would consequently expect to find extremely 
variable discharges, such as have been shown by the 
experiments of Sauvage & Pulin. The increase of the 
discharge with reference to that taking place if there 
were no decrement of temperature, forms an indication 
of the amount of the decrement in temperature. 

As this decrement of temperature increases, the 
flow tends to approach more and more that of a simple 
liquid and does not differ at all when the initial tem- 
perature of the water T —t is equal to or less than 
that of the vapor corresponding to the pressure P t in the 
exhaust space. The discharge will then reach the value 



4 



2g . P ° Pl 



D 

per square centimetre per second. The actual dis- 



72 PLOW OF HOT WATER THROUGH NOZZLES. 

charge is always found to lie between this last value 
and that which we have previously calculated for the 
case when the water is initially at the temperature T . 
For example, for an initial pressure of 6 kilogrammes 
per square centimetre and discharging into the atmos- 
phere, the discharge per second per square centimetre 
should lie between 3130 grammes and 344 grammes, 
the figure just given. In their experiments Sauvage 
& Pulin found discharges for a pressure of 6 kilo- 
grammes per square centimetre, varying irregularly 
around the value 1350 grammes, which would appear 
to indicate that under the conditions of the experi- 
ments the water was at a temperature about 6 degrees 
lower than that of steam at the same pressure. 

For the sake of example I have given in Fig. 4 
nozzle areas for the discharge of hot water from an 
initial pressure of 6 kilogrammes per square centimetre 
with differences of temperature increasing by single 
degrees. The total fall of temperature (t+d) is plotted 
as abscissae, while the ordinates indicate the area of 
flow in square centimetres for a discharge of 1 kilo- 
gramme per second. The curve AB represents the 
case of cold water (without evaporation); CMD that 
of hot water initially at the temperature of vaporization 
(where £ = 0); curves 3, 4, ... 9 correspond to the 
cases where the decrement of temperature is 1°, 2°, etc. 

For a decrement of 2°, for example, the water at 
first flows without causing evaporation and the areas 
decrease up to the point C; then, because evaporation 
takes place, the "area increases, from which it will be 



FLOW OF HOT WATER THROUGH NOZZLES. 73 

seen that if the nozzle is convergent hot water will 
be discharged without any portion being vaporized, at 
its initial temperature and at the pressure correspond- 
ing to that temperature for saturated steam. 

This is not quite general, however. If the curve of 
areas be drawn for decrements of temperature less than 
about 1 degree it will be found that two successive 
minima are obtained, as, for example, curve 2. The 
first minimum a is an angular point; the other occurs 
in the neighborhood of a point M on the curve CD. 
Further, when t is less than 0.395 degree, the first 
minimum is larger than the second, as will be seen 
from curve 1. It would appear from the singular 
form of this curve of areas that the flow in a converg- 
ent nozzle ought to undergo a discontinuity because it 
is impossible for the fluid to be in the states correspond- 
ing to the part of the curve between the points m and n. 
Since the areas of the nozzle are always decreasing, 
there must be a sudden jump from m to n; that is to 
say, an instantaneous partial evaporation, and, which 
is very curious, an equally instantaneous increase in 
the velocity. 

This remarkable phenomenon of a sudden variation 
in the evaporation and in the speed demands very 
careful examination. It would appear incompatible 
a priori with the laws of inertia of material bodies; but 
on thinking of the matter one can see, I think, that these 
discontinuities are not impossible when fluid bodies are 
being experimented with, which can be decomposed 



74 PLOW OF HOT WATER THROUGH XOZZLES. 

actually and not only theoretically into very small 
elementary masses. 

I will conclude with these remarks, which are suffi- 
cient to show that the flow of a hot liquid presents 
peculiarities very well worthy of attention. 



APPENDIX. 



To those wishing to study this subject further, the 
following bibliography may be of assistance: 

-Youxg, Phil. Trans., 1800, p. 187. 
Lagerbein. Amiales de Physique et de Chimie, XVI, 1811 to 

1815, p. 204. 
Schmidt. Poggendorf' s Annalen, II, 1824. 
Banks. Memoirs of the Lit. and Phil. Soc. of Manchester 

(England), Vol. I, p. 389. 
D'Aubissox. Annales des Mines, 1826. 
Koch and Buff. Pog. Ann., 1836-7. 
St. Tenant and Wantzell. Jour, de l'Ecole Polytechnique, 1838; 

Comptes Rendus, 1839-45. 
Tremery. Ann. des Mines, 1844. 

BecquErel and Poxcelet. Comptes Rendus, XXI, 1845. 
Graham. Phil. Trans., 1846, 1849, 1863. 
Froude. Proc. Inst. Civil Eng., VI, 1847. 
Joule and Thomson. Proc. Royal Soc. London, 1856. 
Weisbach. Civil Ingenieur, V, 1859; XII, 1866. 
Joule. Mem. Lit. and Phil. Soc. Manchester, 3d Series, I, p. 102. 
Napier. On the Velocity of Steam and other Gases. Spon, 1866. 
Herrmax. Z. V. Deut. Ing., 1867, p. 345. 
Raxkixe. Outflow of Steam. Van Nostrand's Eng. Mag., Vol. II, 

p. 116. 
Zeuxer. Civil Ing., XX, 1874, and Thermodynamics. 
Fliegxer. Civil Ing., XX, 1874; XXIII, 1874. 
Brovxlee. Trans. Engrs. and Shipbuilders in Scotland, Vol. XVII. 

1874-5. p. 13. 

75 



76 APPENDIX. 

Grashof. Theor. Masch. Leipsic, 1875. 

Hirn. Comptes Rendus, 1886. 

Reynolds. Phil. Mag., 1886. 

Peabody and Kunhardt, Trans. A. S. M. E., X, 1888; XI, 1889. 

Sauvage and Pulin. Ann. des Mines, 1892, p 192. 

De Parenty. Ann. de Chim. et de Phys., 1896-7. 

Unwin. Enc. Brit., 9th Ed. ; Art. Hydromechanics. 

Minary and Resal. Ann. des Mines, 5th Series, Vol. IX. 

Miller and Read. Tech. Quart., Boston, Vol. VIII. 

Kneass. Discharge of Steam through Orifices, Proc. Eng. Club, 

Phila., Vol. VIII, 3, July, 1891. 
.Stodola. Z. V. Deut. Ing., Jan. 10, 1903. 
Borsody and Cairncross. Pressures and Temperatures in Free 

Expansion, Trans. A. S. M. E., Vol. XXVI. 



PLATE L 




EXPERIMENTS ON THE FLOW OF STEAM 




DIAGRAMMATIC ARRANGEMENT OF APPARATUS 



PLATE TT. 



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9 10 11 12 



ORIFICE IN THIN PLATE 



EXPERIMENTS ON THE FLOW OF STEAM 



PLATE TT. 



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A RESAL'S EXPERIMENT 








































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PLATE III. 



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NTS ON THE FLOW OF STEAM 



X CONVERGENT NOZZLE 10.49 MM. Dl AM. 

® " " 15.19 " 

• << " 24.20 " 

+ THIN PLATE ORIFICE 20.12 " 



0.95 



0.55 



0.5 



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0.4 



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>l THE FLOW OF AIR, 



4 SQ. CM". COEFFICIENT 0.966TO 0.985 



0.980 TO 0.991 

'" 0.880 TO 0.920 

0.633 TO 0,845 

0.G33 TO 0.859 



0.95 



0.9 



0.5 



0,45 



0.4 



PLATE IV 




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23 Murray Street and 2j Warren Street \ New York. 



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SCHUMANN, F. A Manual of Heating and Ventilation 

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SCIENCE SERIES, The Van Nostrand. (Follows end of 

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58 D. VAN NOSTRAND COMPANY'S 

WILLIAMSON, R. S. On the Use of the Barometer on 

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60 



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SCIENTIFIC PUBLICATIONS. 63 

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terpretation. Translated from the French of M. Argand by 
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Eleventh American edition. 

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No. 55. SEWER GASES: THEIR NATURE AND ORIGIN. By A. 

de Varona. Second edition, revised and enlarged. 

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SCIENTIFIC PUBLICATIONS. 65 

No. 87. TREATISE ON THE THEORY OF THE CONSTRUCTION 

of Helicoidal Oblique Arches. By John L. Culley, C.E. 

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66 D. VAN NOSTRAND COMPANY'S 

No. 103. THE MICROSCOPICAL EXAMINATION OF POTABLE 

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No. 107. A GRAPHICAL METHOD FOR SWING BRIDGES. A 

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La Rue. 

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No. no. TRANSITION CURVES. A Field-book for Engineers, Con- 
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No. in. GAS-LIGHTING AND GAS-FITTING. Specifications and 
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No. 112. A PRIMER ON THE CALCULUS. By E. Sherman Gould, 
M. Am. Soc. C. E. Third edition, revised and enlarged. 

No. 113. PHYSICAL PROBLEMS and Their Solution. By A. Bour- 
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No. 114. MANUAL OF THE SLIDE RULE. By F. A. Halsey, of 

the "American Machinist." Third edition, corrected. 

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from Scribner's Pocket Table Book.) 



MAR 3 19C5 



